WEBVTT

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Game Basics
The Fundamental Mathematics of Games: Vectors and Linear Independence

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Hello, everyone

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This is Lee Deok-woo, from Game Mathematics

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In this session, as explained in the previous session

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we will delve into vectors

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which form the foundation of game systems

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I will cover the definition of vectors, operations involving vectors

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and the axioms that constitute vector spaces

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Then, we will explore linear independence and dimensions

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The discussion is divided into two parts

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First is understanding the generation system of vectors

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within vector spaces

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And next is exploring the mathematical concepts of basis and dimensions

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Vectors

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Let’s begin by explaining the definition of vectors

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and understanding what vectors really are

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In the previous session, we learned about numbers

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Numbers are simply sets of elements

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that can be visualized

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as a line of points arranged seamlessly

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Additionally, numbers are not just sets

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based on axioms

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they are governed by binary operations

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allowing new numbers to be generated

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However, when we consider the visual representation of numbers

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they are limited to being

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expressed as points on a straight line

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When we want to represent something through a point on a straight line

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this limitation becomes clear

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To depict anything of visual significance

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we need at least a plane

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where we can draw

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and manipulate various shapes

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Using only numbers confined to a straight line

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makes it challenging to express much

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To address this, we need to extend numbers 
beyond one-dimensional space

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If the domain of numbers is considered one-dimensional

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we can say that we need to extend it to a two-dimensional domain

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Let’s explore how this extension is achieved

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In mathematics, the two-dimensional 
representation method commonly used

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is where the x-axis points to the right and the y-axis points upward

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and we usually call this a graph

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This representation is widely known as

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the Cartesian Coordinate System

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In English, it is called the Cartesian Coordinate System

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The Cartesian coordinate system is created

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by taking a straight line, representing numbers

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and intersecting it perpendicularly with another straight line

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to form a plane

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This perpendicular intersection forms

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the basis of the Cartesian coordinate system

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In this system, the positive direction

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commonly called the plus direction

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points to the right and upward

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As shown in the diagram, the rightward and upward directions

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are defined as independent sets

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of real numbers

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which intersect at 90 degrees

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The perpendicular intersection is referred to as "orthogonality"

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which carries a significant meaning

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To briefly explain, orthogonality means

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more than just a 90-degree arrangement

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it signifies that the set of real numbers along the horizontal axis

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and the set of real numbers along the vertical axis are unrelated

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This is the essence of the 90-degree intersection

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By creating this extended plane

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we now have a stage

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to work with

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visually meaningful elements

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From the perspective of numbers as sets

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this plane can be seen as

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the Cartesian product of two sets of real numbers, R x R

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The elements of this Cartesian product

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are expressed using a notation called tuples

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which use parentheses and commas

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if X represents an element of the first set of real numbers

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and Y represents an element of the second set

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represented as an element of the Cartesian product

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as (X, Y)

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This element is used to describe the Cartesian product visually

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and we call this a coordinate

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So we say that a coordinate is used to describe the Cartesian product

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These coordinates are typically represented

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when visualized

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as points on the plane

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where the x-axis and y-axis intersect perpendicularly

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As we discussed earlier, numbers can be represented in two ways

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as points at specific locations

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or as arrows

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emanating from the origin

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Thus, an element of this two-dimensional space

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the Cartesian product of two sets of real numbers

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can be represented as a point or as an arrow

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originating from the origin

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The choice between these two representations

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as a point or as an arrow

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depends on the context

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When we discuss affine spaces later, we’ll clarify when to use what

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For now, it suffices to understand that

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both representations are possible

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Now, if we revisit the straight line

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represented as a set of numbers

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and the plane expressed through Cartesian products

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from the perspective of sets and elements

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In the case of a straight line, it is represented by a single element

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from the set of real numbers

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Now, with ordered pairs as elements of the Cartesian product

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expressed as x, y, we can represent elements on the plane

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This establishes the basis for representing elements

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So, what do we call this set?

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We can call it a Cartesian product

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or refer to it as elements of the Cartesian product

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However, describing it as a Cartesian product of real numbers

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or elements of a Cartesian product of real numbers

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is quite long and difficult

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This is because

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when we use computer graphics in the future

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it becomes the foundation for everything

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Such complex explanations are inconvenient

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That is why we decided to give these sets and elements

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distinct names to use and call them by

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Those names are vectors and vector spaces

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As shown in the diagram

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we define vectors and vector spaces separately

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In the two-dimensional plane

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and extending further to three dimensions and beyond

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the concepts of numbers and sets of numbers

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are generalized into vectors and vector spaces

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This can be viewed as an extension

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As I mentioned at the end of Lecture 1

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analyzing and representing the structure of number sets

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from a broader perspective is more versatile

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It allows for broader applications

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Therefore, vector and vector space refer to

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number sets

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which in this context use real numbers

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but from the perspective of fields

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they form a Cartesian product

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Understanding this from a broader viewpoint is helpful

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Thus, vector spaces

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can be seen as Cartesian products of two fields

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from the perspective of sets

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The elements of these fields are called scalars

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Combining these scalars into ordered pairs

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defines a vector

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This structural perspective

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enables the definition of vector spaces and vectors

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The vector space is a concept of sets

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and vectors are its elements

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This provides a systematic framework

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If we think of them as simple sets

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vectors merely represent static objects

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They are just elements within sets

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However, to represent something dynamic over time

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like forming a system that creates animations

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the system of vector spaces

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requires binary operations similar to number sets

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These are called the fundamental operations of vectors

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Regarding the fundamental operations of vectors

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I’ve summarized two main types

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The first is vector addition

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The concept of a vector

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initially begins as the Cartesian product of two sets

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but adding another field results in

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a Cartesian product of three, four, and so on

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This can be extended indefinitely

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To simplify

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let’s consider the Cartesian product of two sets

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Here, an element can be expressed as the ordered pair a, b

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If another element is c, d

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vector addition is defined as follows

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Add the corresponding components of each pair

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This independent addition forms vector addition

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When this vector addition is visualized

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it appears like this

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For example, adding (1,2) and (3,1)

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Add the x-values 1 + 3

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and add the y-values 2 + 1

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This gives the vector addition result

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How can I explain this?

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If I compare this mechanism

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to real-world substances

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It’s like water and oil

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Since x and y are perpendicular

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we said they are unrelated, right?

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They cannot mix

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They cannot influence each other

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A large x-value does not affect y

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and no matter how big or small y becomes

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it does not influence x

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These are two independent sets

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This can be compared to water and oil

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Now, let’s consider (1,2)

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as 10 ml of water and 20 ml of oil

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and (3, 2) as 30 ml of water and 10 ml of oil

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If you combine them in a cup, what would happen?

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Obviously, water and oil would remain separate

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Thus, vector addition

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is like water and oil along the x-axis

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add x-values and y-values separately

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as they do not influence each other

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and this completes the calculation

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This is the first fundamental operation of vectors

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You might think that the next one is multiplication between vectors

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Your guess is wrong

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It is the multiplication of a vector and a scalar

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As explained earlier, a scalar refers to

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an element of a field set

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We call the elements of a field set scalars, right?

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If we use the set of real numbers as our field

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then scalars here are real numbers

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Thus, it could be described as multiplication 
between vectors and real numbers

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But calling it the multiplication of vectors and scalars

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provides a broader and more structural perspective

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It offers a clearer understanding of the concept

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So, the multiplication of a vector and a scalar

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is represented as k, where k is a scalar value

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When multiplying a vector (a, b) by k

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you get ka, kb, and reversing the order gives the same result

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Reversing it would give ak, bk

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Here, a and b are both scalars

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and the multiplier k is also a scalar

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so ka, kb, and ak, bk

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may be different or the same

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However, based on the axioms of fields

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we know that multiplication is commutative

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Therefore, ka equals ak

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as defined by the axioms of fields

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Therefore, they produce the same result

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In conclusion, multiplying (a,b) by k has the same result 
as multiplying each component of a,b by k

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This is verified by the axioms of fields

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This multiplication of vectors and scalars

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is adopted as the second fundamental operation

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When graphing this multiplication

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the result is as follows

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When we represent a vector as an arrow originating from the origin

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imagine it visually

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The arrow from the origin has

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what we commonly call a slope

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This slope defines a line that perfectly aligns with the arrow

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It can be seen as a line

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This is essentially an infinite line

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The line follows the slope of the vector

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On this infinite line

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any new vector created by scalar multiplication

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will always lie on the infinite line

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It will always be on this green line

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Thus, scalar multiplication of vectors

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can be seen as a method for creating one-dimensional vectors

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This is because they exist only along the line

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To summarize the two concepts

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First, vector addition

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moves x-values and y-values

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independently of each other

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resulting in a parallel shift by x and y

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This produces a visual representation of parallel movement

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Second, scalar multiplication of a vector

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creates a vector centered on the origin

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The scalar multiplication of that vector

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lies on the infinite line with the same slope

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It exists only along that invisible line

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This is how a vector is generated

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Previously, when we discussed numbers

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we understood positive and negative directions 
originating from the origin

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I explained that understanding it this way helps

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This is the same concept

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Usuing (0,0) as the reference point

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it encompasses both positive and negative directions

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On a line with a specific slope

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a system generates new vectors

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This can be seen as a system for creating one-dimensional vectors

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These are the two fundamental operations

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00:17:28.142 --> 00:17:30.446
In practical applications

274
00:17:30.446 --> 00:17:33.610
these two operations are not commonly used directly

275
00:17:33.610 --> 00:17:35.610
You’ve likely heard of them

276
00:17:35.610 --> 00:17:39.096
dot products and cross products of vectors

277
00:17:39.096 --> 00:17:42.000
These are widely used in practical applications

278
00:17:42.000 --> 00:17:45.829
They are operations meant specifically for applications

279
00:17:45.829 --> 00:17:47.230
We will learn about them later

280
00:17:47.230 --> 00:17:50.585
At this stage, as we learn the definition of vectors

281
00:17:50.585 --> 00:17:52.880
we focus on these two fundamental operations

282
00:17:52.880 --> 00:17:56.135
Consider this as the focus of our current learning

283
00:17:58.719 --> 00:18:02.374
Another important concept in vectors is their magnitude

284
00:18:02.374 --> 00:18:06.120
Previously, we discussed the magnitude of numbers

285
00:18:06.120 --> 00:18:10.297
as the shortest distance from the origin

286
00:18:10.297 --> 00:18:11.485
The same applies to vectors

287
00:18:11.485 --> 00:18:14.658
The shortest distance from the origin

288
00:18:14.658 --> 00:18:17.329
is defined as the magnitude of a vector

289
00:18:17.329 --> 00:18:19.815
This can be referred to as the distance from the origin

290
00:18:19.815 --> 00:18:21.424
the length from the origin

291
00:18:21.424 --> 00:18:23.459
the vector’s length

292
00:18:23.459 --> 00:18:26.609
or in mathematical terms, the norm

293
00:18:26.609 --> 00:18:30.927
These three terms essentially mean the same

294
00:18:30.927 --> 00:18:34.283
To calculate the magnitude or norm

295
00:18:34.283 --> 00:18:38.910
it also represents the shortest distance from the origin

296
00:18:38.910 --> 00:18:42.530
In a plane, the shortest distance from the origin

297
00:18:42.530 --> 00:18:44.875
is, as shown in the diagram

298
00:18:44.875 --> 00:18:47.526
a right triangle

299
00:18:47.526 --> 00:18:52.289
It is the distance of the hypotenuse of the right triangle

300
00:18:52.289 --> 00:18:54.344
So, how can we calculate this distance?

301
00:18:54.344 --> 00:18:57.520
You are probably familiar with the Pythagorean theorem

302
00:18:57.520 --> 00:19:02.919
It states that the square of the base plus the square of the height

303
00:19:02.919 --> 00:19:04.569
equals the square of the hypotenuse

304
00:19:04.569 --> 00:19:08.160
this is expressed as a^2 + b^2 = c^2

305
00:19:08.160 --> 00:19:10.032
Using this

306
00:19:10.032 --> 00:19:12.846
we can calculate the shortest distance from the origin

307
00:19:12.846 --> 00:19:15.557
This distance is

308
00:19:15.557 --> 00:19:18.400
the square root of x^2 + y^2

309
00:19:18.400 --> 00:19:21.141
Thus, this shortest distance, as we discussed earlier

310
00:19:21.141 --> 00:19:24.651
is similar to finding the shortest distance for a number

311
00:19:24.651 --> 00:19:26.169
by using absolute value symbols

312
00:19:26.169 --> 00:19:27.940
We denote this by placing vertical bars around the number

313
00:19:27.940 --> 00:19:29.654
By using absolute value symbols

314
00:19:29.654 --> 00:19:32.070
we can find the shortest distance

315
00:19:32.070 --> 00:19:34.303
Similarly, for vectors

316
00:19:34.303 --> 00:19:37.120
if we place vertical bars around a vector

317
00:19:37.120 --> 00:19:39.389
it represents the shortest distance from the origin

318
00:19:39.389 --> 00:19:41.378
or the magnitude of the vector

319
00:19:41.378 --> 00:19:46.274
Generally, using double vertical bars

320
00:19:46.274 --> 00:19:48.359
is more commonly used

321
00:19:48.359 --> 00:19:51.248
In this course, however

322
00:19:51.248 --> 00:19:54.350
to simplify the expressions

323
00:19:54.350 --> 00:19:57.880
we will use only a single vertical bar

324
00:19:57.880 --> 00:20:00.545
Using double bars can make the notation cumbersome

325
00:20:00.545 --> 00:20:02.504
To ensure consistency in meaning

326
00:20:02.504 --> 00:20:06.199
we will use a single vertical bar to denote magnitude

327
00:20:06.199 --> 00:20:09.257
Like this

328
00:20:09.257 --> 00:20:12.432
Once we have determined the magnitude of a vector

329
00:20:12.432 --> 00:20:14.909
the next concept to understand is

330
00:20:14.909 --> 00:20:16.720
the unit vector

331
00:20:16.720 --> 00:20:21.348
A unit vector is a vector with a magnitude of 1

332
00:20:21.348 --> 00:20:25.590
To find a unit vector

333
00:20:25.590 --> 00:20:28.317
consider the original magnitude of the vector

334
00:20:28.317 --> 00:20:33.348
By multiplying this magnitude by its reciprocal

335
00:20:33.348 --> 00:20:36.560
the multiplicative inverse

336
00:20:36.560 --> 00:20:38.317
the magnitude becomes 1

337
00:20:38.317 --> 00:20:39.759
This results in the multiplicative identity

338
00:20:39.759 --> 00:20:43.283
When you multiply a number by its inverse

339
00:20:43.283 --> 00:20:45.580
you get the identity element 1

340
00:20:45.580 --> 00:20:47.710
The same principle applies here

341
00:20:47.710 --> 00:20:50.802
That covers the explanation of vector magnitude

342
00:20:50.802 --> 00:20:55.484
Finally, based on these fundamental elements

343
00:20:55.484 --> 00:20:57.875
let’s explore how the vector system

344
00:20:57.875 --> 00:21:02.956
is structured

345
00:21:02.956 --> 00:21:05.771
As mentioned briefly earlier

346
00:21:05.771 --> 00:21:10.227
a vector, from the perspective of sets

347
00:21:10.227 --> 00:21:13.760
can be seen as a collection of points

348
00:21:13.760 --> 00:21:17.293
However, if we view it as a closed system

349
00:21:17.293 --> 00:21:19.877
a closed system able to generate

350
00:21:19.877 --> 00:21:23.450
new elements

351
00:21:23.450 --> 00:21:25.123
then we need to organize the operations

352
00:21:25.123 --> 00:21:28.986
The two operations mentioned earlier—vector addition

353
00:21:28.986 --> 00:21:33.578
and vector-scalar multiplication

354
00:21:33.578 --> 00:21:35.402
must exist to enable the creation

355
00:21:35.402 --> 00:21:40.188
of new vectors within the system

356
00:21:40.188 --> 00:21:45.334
These operations aren’t just arbitrarily defined

357
00:21:45.334 --> 00:21:49.560
As with number sets

358
00:21:49.560 --> 00:21:54.058
we organize axioms for operations within a field

359
00:21:54.058 --> 00:21:56.830
If these axioms are satisfied

360
00:21:56.830 --> 00:22:00.070
we can define a system accordingly

361
00:22:00.070 --> 00:22:03.427
This allows us to structurally

362
00:22:03.427 --> 00:22:07.767
view the vector space system

363
00:22:07.767 --> 00:22:10.249
It provides the foundation for this perspective

364
00:22:10.249 --> 00:22:12.752
Thus, a vector space

365
00:22:12.752 --> 00:22:15.639
is a system encompassing vectors

366
00:22:15.639 --> 00:22:18.849
The broad concept of a vector space

367
00:22:18.849 --> 00:22:21.315
is also governed

368
00:22:21.315 --> 00:22:24.661
by several axioms

369
00:22:24.661 --> 00:22:29.158
It is composed of 8 axioms

370
00:22:29.158 --> 00:22:32.267
Let’s examine these axioms

371
00:22:32.267 --> 00:22:36.706
Here, u, v, and w represent vectors

372
00:22:36.706 --> 00:22:39.090
and a and b represent scalars

373
00:22:39.090 --> 00:22:42.883
The first axiom is the associative property of addition

374
00:22:42.883 --> 00:22:47.440
Addition refers to vector addition

375
00:22:47.440 --> 00:22:49.031
The associative property means that

376
00:22:49.031 --> 00:22:53.040
the grouping of additions doesn’t affect the result

377
00:22:53.040 --> 00:22:56.810
For example, with vectors u, v, and w

378
00:22:56.810 --> 00:22:59.920
adding v and w first, and then u

379
00:22:59.920 --> 00:23:02.273
or adding u and v first, then w

380
00:23:02.273 --> 00:23:03.841
produces the same result

381
00:23:03.841 --> 00:23:09.149
This is because each component of a vector is a scalar

382
00:23:09.149 --> 00:23:13.627
The x-components are added separately from the y-components

383
00:23:13.627 --> 00:23:18.737
Since x and y are elements of a field

384
00:23:18.737 --> 00:23:21.830
the associative property naturally holds

385
00:23:21.830 --> 00:23:24.205
Thus, even for vectors

386
00:23:24.205 --> 00:23:28.331
this property holds true

387
00:23:28.331 --> 00:23:31.629
The first axiom of associativity is satisfied

388
00:23:31.629 --> 00:23:34.725
The second is the commutative property of addition

389
00:23:34.725 --> 00:23:38.994
This, too, is inherent in vector systems based on fields

390
00:23:38.994 --> 00:23:43.080
Naturally, it must hold true

391
00:23:43.080 --> 00:23:45.959
The third is the existence of an additive identity

392
00:23:45.959 --> 00:23:52.690
This refers to the zero vector, where all components are zero

393
00:23:52.690 --> 00:23:57.558
Adding 0 to each component leaves the vector unchanged

394
00:23:57.558 --> 00:23:59.947
so the vector's value remains the same

395
00:23:59.947 --> 00:24:02.028
Therefore, in vector addition

396
00:24:02.028 --> 00:24:05.397
the additive identity in a vector space

397
00:24:05.397 --> 00:24:08.980
is defined as the zero vector

398
00:24:08.980 --> 00:24:11.160
From here, we can define the additive inverse

399
00:24:11.160 --> 00:24:17.330
The additive inverse of a vector is

400
00:24:17.330 --> 00:24:21.311
the vector obtained by negating it

401
00:24:21.311 --> 00:24:25.540
This represents a vector pointing in the opposite direction

402
00:24:25.540 --> 00:24:27.711
so the additive inverse is

403
00:24:27.711 --> 00:24:31.455
denoted as -v

404
00:24:31.455 --> 00:24:34.079
Now, let’s move on to multiplication

405
00:24:34.079 --> 00:24:38.081
Multiplication here does not involve 
vectors multiplying each other

406
00:24:38.081 --> 00:24:42.498
Instead, it is the multiplication of a vector by a scalar, 
which might feel a bit different

407
00:24:42.498 --> 00:24:45.193
This is because vectors

408
00:24:45.193 --> 00:24:50.920
and vector spaces are not solely composed of vectors

409
00:24:50.920 --> 00:24:52.983
As mentioned earlier, the foundation

410
00:24:52.983 --> 00:24:57.530
that forms a vector space lies

411
00:24:57.530 --> 00:25:00.391
in the structure of a field set

412
00:25:00.391 --> 00:25:04.560
Thus, vector spaces operate over a field

413
00:25:04.560 --> 00:25:07.095
and are built on this field

414
00:25:07.095 --> 00:25:09.686
As a result, scalars

415
00:25:09.686 --> 00:25:12.530
must always be considered

416
00:25:12.530 --> 00:25:15.765
This is a fundamental difference

417
00:25:15.765 --> 00:25:18.079
between number sets and vectors

418
00:25:18.079 --> 00:25:21.221
Scalar multiplication involves compatibility

419
00:25:21.221 --> 00:25:25.186
a property known as compatibility in English

420
00:25:25.186 --> 00:25:28.588
No matter which scalar is used

421
00:25:28.588 --> 00:25:31.766
changing the order does not alter the result

422
00:25:31.766 --> 00:25:36.550
This is because it satisfies the commutative 
and associative properties of multiplication

423
00:25:36.550 --> 00:25:41.318
Hence, it naturally holds true

424
00:25:41.318 --> 00:25:45.904
When multiplying a scalar by 1

425
00:25:45.904 --> 00:25:47.808
each component is multiplied by 1

426
00:25:47.808 --> 00:25:49.882
This is due to the multiplicative identity

427
00:25:49.882 --> 00:25:52.531
so it results in the same vector

428
00:25:52.531 --> 00:25:55.008
Thus, the identity for scalar multiplication

429
00:25:55.008 --> 00:26:00.354
is defined as 1

430
00:26:00.354 --> 00:26:03.319
Next, when both addition and multiplication

431
00:26:03.319 --> 00:26:06.062
are used together

432
00:26:06.062 --> 00:26:09.359
we previously defined this as the distributive property

433
00:26:09.359 --> 00:26:12.404
When working with vectors

434
00:26:12.404 --> 00:26:15.789
we deal with both vectors and scalars

435
00:26:15.789 --> 00:26:20.319
If the elements in addition are vectors

436
00:26:20.319 --> 00:26:24.400
it results in a form like a(u+v)

437
00:26:24.400 --> 00:26:30.392
and this is equivalent to multiplying 
each vector by the scalar individually

438
00:26:30.392 --> 00:26:32.759
Thus, the distributive property holds

439
00:26:32.759 --> 00:26:36.106
If the elements in addition

440
00:26:36.106 --> 00:26:38.506
are scalars instead of vectors

441
00:26:38.506 --> 00:26:40.479
the result is still the same

442
00:26:40.479 --> 00:26:44.880
It equals the sum of the products of the scalars

443
00:26:44.880 --> 00:26:49.160
This indicates that the distributive property has two variations

444
00:26:49.160 --> 00:26:53.261
With this, we have covered all 8 axioms

445
00:26:53.261 --> 00:26:55.508
that define the system of vector spaces

446
00:26:55.508 --> 00:27:00.874
Finally, the definition of a vector space

447
00:27:00.874 --> 00:27:05.192
is a system that extends the elements of a field set

448
00:27:05.192 --> 00:27:09.349
by combining them into ordered pairs

449
00:27:09.349 --> 00:27:11.443
This is the expanded concept

450
00:27:11.443 --> 00:27:14.279
When using this practically

451
00:27:14.279 --> 00:27:16.239
we need to specify the number set

452
00:27:16.239 --> 00:27:19.127
The most commonly used set is

453
00:27:19.127 --> 00:27:21.650
the set of real numbers

454
00:27:21.650 --> 00:27:26.430
Thus, when using vector spaces, 
we often use the set of real numbers

455
00:27:26.430 --> 00:27:28.735
To build a vector space

456
00:27:28.735 --> 00:27:33.982
requires specific actions in applications

457
00:27:33.982 --> 00:27:36.857
A system based on real numbers

458
00:27:36.857 --> 00:27:40.054
is called

459
00:27:40.054 --> 00:27:44.680
a real vector space

460
00:27:44.680 --> 00:27:50.360
This serves as a practical stage for various applications

461
00:27:50.360 --> 00:27:53.586
From now on, in the broader context of vector spaces

462
00:27:53.586 --> 00:27:55.981
we will use

463
00:27:55.981 --> 00:27:59.036
real vector spaces

464
00:27:59.036 --> 00:28:02.582
to represent objects on a plane

465
00:28:02.582 --> 00:28:05.706
This summarizes the direction of future discussions

466
00:28:06.310 --> 00:28:10.271
Linear Independence

467
00:28:10.271 --> 00:28:16.490
In this session, we will explore 
the concepts of linear independence and dimension

468
00:28:16.490 --> 00:28:21.352
Today's discussion consists of two parts

469
00:28:21.352 --> 00:28:25.520
First, we will examine the system for generating vectors

470
00:28:25.520 --> 00:28:29.415
within a vector space

471
00:28:29.415 --> 00:28:32.400
Next, we will explore the mathematical concepts

472
00:28:32.400 --> 00:28:36.379
of basis and dimension

473
00:28:36.379 --> 00:28:41.096
Let’s start with the vector generation system

474
00:28:41.096 --> 00:28:44.295
A vector generation system

475
00:28:44.295 --> 00:28:46.597
uses the fundamental operations

476
00:28:46.597 --> 00:28:49.230
of vectors discussed earlier

477
00:28:49.230 --> 00:28:53.822
vector addition and vector-scalar multiplication

478
00:28:53.822 --> 00:28:56.514
By applying these two operations

479
00:28:56.514 --> 00:29:01.009
we create new vectors in the system

480
00:29:01.009 --> 00:29:05.895
This is similar to the number system 
discussed in the first session

481
00:29:05.895 --> 00:29:09.291
where numbers are combined through binary operations

482
00:29:09.291 --> 00:29:12.238
to generate new numbers

483
00:29:12.238 --> 00:29:15.008
It works in a similar manner

484
00:29:15.008 --> 00:29:17.650
This generation system

485
00:29:17.650 --> 00:29:20.745
is called a linear combination

486
00:29:20.745 --> 00:29:23.800
In mathematics, creating new vectors this way

487
00:29:23.800 --> 00:29:26.324
is referred to as span

488
00:29:26.324 --> 00:29:27.351
It does not use the term 'create'

489
00:29:27.351 --> 00:29:31.769
but instead uses 'span'

490
00:29:31.769 --> 00:29:34.256
Let’s explore how spanning works

491
00:29:34.256 --> 00:29:36.810
and examine the relevant equations

492
00:29:36.810 --> 00:29:42.955
As shown here, by multiplying a scalar a with a vector v

493
00:29:42.955 --> 00:29:46.119
and summing these scalar-vector products

494
00:29:46.119 --> 00:29:50.113
you can combine the elements

495
00:29:50.113 --> 00:29:52.236
The result is

496
00:29:52.236 --> 00:29:55.764
a vector because a scalar times a vector is still a vector

497
00:29:55.764 --> 00:29:58.999
Adding vectors also results in a vector

498
00:29:58.999 --> 00:30:01.870
Thus, a new vector is generated

499
00:30:01.870 --> 00:30:06.470
If the resulting vector is denoted as v'

500
00:30:06.470 --> 00:30:11.129
we can define the general formula for generating vectors like this

501
00:30:11.129 --> 00:30:13.803
So, this n

502
00:30:13.803 --> 00:30:17.386
allows us to sun as many terms as needed

503
00:30:17.386 --> 00:30:19.811
By adding as much as necessary

504
00:30:19.811 --> 00:30:21.482
we can generate new vectors

505
00:30:21.482 --> 00:30:25.319
You can think of it this way

506
00:30:25.319 --> 00:30:29.301
Now, there are two important concepts to understand here

507
00:30:29.301 --> 00:30:33.760
They are linear dependence and linear independence

508
00:30:33.760 --> 00:30:36.280
So, when two vectors

509
00:30:36.280 --> 00:30:39.718
or n vectors are dependent on one another

510
00:30:39.718 --> 00:30:41.575
or independent of each other

511
00:30:41.575 --> 00:30:42.850
we say they are independent

512
00:30:42.850 --> 00:30:47.478
These are the two key concepts

513
00:30:47.478 --> 00:30:52.123
Rather than vaguely saying they overlap

514
00:30:52.123 --> 00:30:53.358
or that they are separate

515
00:30:53.358 --> 00:30:55.760
we shouldn't explain it like this

516
00:30:55.760 --> 00:31:00.265
but we must define dependence 
and independence mathematically

517
00:31:00.265 --> 00:31:02.995
clearly using equations

518
00:31:02.995 --> 00:31:07.790
In mathematics, linear dependence is defined as follows

519
00:31:07.790 --> 00:31:10.941
If a linear combination of vectors satisfies this equation

520
00:31:10.941 --> 00:31:14.344
and results in the zero vector

521
00:31:14.344 --> 00:31:16.463
then we have linear dependence

522
00:31:16.463 --> 00:31:20.116
In any linear combination that produces the zero vector

523
00:31:20.116 --> 00:31:27.181
if any of the coefficients a1, a2, a3, ..., an

524
00:31:27.181 --> 00:31:31.280
are non-zero

525
00:31:31.280 --> 00:31:35.408
it is defined as linear dependence

526
00:31:35.408 --> 00:31:39.160
This is the formal definition of linear dependence

527
00:31:39.160 --> 00:31:43.500
If all a's are non-zero

528
00:31:43.500 --> 00:31:45.947
it is a case of linear dependence

529
00:31:45.947 --> 00:31:49.551
The definition of linear independence is similar but different

530
00:31:49.551 --> 00:31:53.898
If a linear combination produces the zero vector

531
00:31:53.898 --> 00:31:57.872
but at least one coefficient is zero

532
00:31:57.872 --> 00:32:02.737
not all coefficients need to be non-zero

533
00:32:02.737 --> 00:32:06.539
The vectors in such an equation are linearly independent

534
00:32:06.539 --> 00:32:10.629
This implies that is even one is 0, 
the linear independence does not hold

535
00:32:10.629 --> 00:32:14.442
All coefficients must not be zero

536
00:32:14.442 --> 00:32:17.038
If any coefficient is 0

537
00:32:17.038 --> 00:32:20.670
it breaks the condition of linear independence

538
00:32:20.670 --> 00:32:22.810
This is how it works

539
00:32:22.810 --> 00:32:29.740
Now, you might wonder what these terms really mean

540
00:32:29.740 --> 00:32:32.705
It may not feel intuitive yet

541
00:32:32.705 --> 00:32:35.745
Let’s look at two simple examples

542
00:32:35.745 --> 00:32:40.036
to examine linear combinations

543
00:32:40.036 --> 00:32:43.194
and understand the relationship 
between dependence and independence

544
00:32:43.194 --> 00:32:45.009
We’ll explore this in detail

545
00:32:45.009 --> 00:32:47.839
I’ve prepared two problems for us

546
00:32:47.839 --> 00:32:53.149
Are the vectors (1,1) and (2,2) linearly 
dependent or independent?

547
00:32:53.149 --> 00:32:55.275
Here is the question

548
00:32:55.275 --> 00:32:56.807
Let’s assign coefficients a1

549
00:32:56.807 --> 00:33:00.639
to (1,1) and a2 to (2,2)

550
00:33:00.639 --> 00:33:04.068
The coefficients a1 and a2 are scalars

551
00:33:04.068 --> 00:33:06.618
If these coefficients produce the zero vector

552
00:33:06.618 --> 00:33:12.938
and if a1 and a2 are non-zero

553
00:33:12.938 --> 00:33:15.536
this indicates linear dependence

554
00:33:15.536 --> 00:33:18.667
For example, if a1 = 2

555
00:33:18.667 --> 00:33:22.119
and a2 = -2, the result is the zero vector

556
00:33:22.119 --> 00:33:25.799
Thus, we say these vectors

557
00:33:25.799 --> 00:33:28.402
are linearly dependent

558
00:33:28.402 --> 00:33:29.745
Now, let’s consider (1,2)

559
00:33:29.745 --> 00:33:31.570
in a different example

560
00:33:31.570 --> 00:33:35.334
Are (1,2) and (2,1) dependent or independent?

561
00:33:35.334 --> 00:33:37.763
To solve this problem

562
00:33:37.763 --> 00:33:43.645
we’ll use scalar multiplication with vectors

563
00:33:43.645 --> 00:33:47.949
and apply the axioms of vector spaces

564
00:33:47.949 --> 00:33:49.994
We can expand the equations

565
00:33:49.994 --> 00:33:52.534
After proceeding step by step

566
00:33:52.534 --> 00:33:56.417
the x-component becomes a1 + 2a2

567
00:33:56.417 --> 00:34:00.396
and the y-component becomes 2a1 + a2

568
00:34:00.396 --> 00:34:04.307
If both components equal zero

569
00:34:04.307 --> 00:34:06.509
we check if a solution exists

570
00:34:06.509 --> 00:34:09.915
If a1 and a2 satisfy both equations

571
00:34:09.915 --> 00:34:13.433
this determines their relationship

572
00:34:13.433 --> 00:34:16.093
The only solution is

573
00:34:16.093 --> 00:34:18.649
a1=0 and a2=0

574
00:34:18.649 --> 00:34:23.218
Thus, the vectors (1,2) and (2,1)

575
00:34:23.218 --> 00:34:29.600
are linearly independent

576
00:34:29.600 --> 00:34:35.312
Now that we’ve defined 
linear dependence and independence

577
00:34:35.312 --> 00:34:37.439
you might wonder why this is important

578
00:34:37.439 --> 00:34:43.603
It is related to the system for generating new vectors

579
00:34:43.603 --> 00:34:48.560
Let’s assume we want to create a vector using linear combinations

580
00:34:48.560 --> 00:34:53.321
Suppose we want to generate the vector (5,5)

581
00:34:53.321 --> 00:34:58.449
We’ll explore how to do this using linear combinations

582
00:34:58.449 --> 00:35:03.117
Let’s assume a1, and a2

583
00:35:03.117 --> 00:35:06.575
are the only coefficients we use

584
00:35:06.575 --> 00:35:13.560
If we combine a1v1 and a2v2 to create the vector

585
00:35:13.560 --> 00:35:16.785
I chose two vectors

586
00:35:16.785 --> 00:35:20.518
(1,0) and (0,1)

587
00:35:20.518 --> 00:35:24.919
Can we use them to linearly combine and form (5,5)?

588
00:35:24.919 --> 00:35:28.523
If we multiply each by 5, as shown

589
00:35:28.523 --> 00:35:32.879
we can generate (5,5) through linear combinations

590
00:35:32.879 --> 00:35:36.092
This might seem too obvious

591
00:35:36.092 --> 00:35:39.576
But are there other pairs that can achieve the same?

592
00:35:39.576 --> 00:35:43.845
For instance, consider the vectors (1,3) and (2,1)

593
00:35:43.845 --> 00:35:46.107
We can combine these two vectors

594
00:35:46.107 --> 00:35:48.466
to create (5,5)

595
00:35:48.466 --> 00:35:52.407
For example, if we mutiply (1,3) by 2

596
00:35:52.407 --> 00:35:56.679
and (2,1) by 1

597
00:35:56.679 --> 00:35:59.818
we can create (5,5) through a linear combination

598
00:36:01.959 --> 00:36:08.189
So, it’s not just these two vectors

599
00:36:08.189 --> 00:36:09.824
you could use other vectors as well

600
00:36:09.824 --> 00:36:12.847
to generate (5,5), right?

601
00:36:12.847 --> 00:36:14.795
You might ask this question

602
00:36:14.795 --> 00:36:18.318
There are many linear combinations of other vectors

603
00:36:18.318 --> 00:36:22.320
that can generate (5,5)

604
00:36:22.320 --> 00:36:28.306
For example, by combining (1,2)

605
00:36:28.306 --> 00:36:31.748
you can generate not just (5,5)

606
00:36:31.748 --> 00:36:33.520
but any vector on the plane

607
00:36:33.520 --> 00:36:35.589
How can we do this?

608
00:36:35.589 --> 00:36:41.560
If we assume an arbitrary point (x,y) on the plane

609
00:36:41.560 --> 00:36:43.652
although we do not know the coefficient

610
00:36:43.652 --> 00:36:47.594
we define a1 and b1 as a and b

611
00:36:47.594 --> 00:36:50.100
Then, the equation becomes

612
00:36:50.100 --> 00:36:52.268
a system of linear equations

613
00:36:52.268 --> 00:36:55.405
where 2a + b= x

614
00:36:55.405 --> 00:36:57.959
and a + 3b = y

615
00:36:57.959 --> 00:37:03.033
This forms a system of equations

616
00:37:03.033 --> 00:37:05.518
Can we solve this system of equations?

617
00:37:05.518 --> 00:37:06.959
Of course, we can

618
00:37:06.959 --> 00:37:09.795
Multiply one equation by 2 and subtract the other

619
00:37:09.795 --> 00:37:13.839
This gives -2b = x - 2y

620
00:37:13.839 --> 00:37:19.439
Dividing by -2, we can find the solution for b

621
00:37:19.439 --> 00:37:21.253
and once b is found

622
00:37:21.253 --> 00:37:23.919
we can calculate a as well

623
00:37:23.919 --> 00:37:26.828
Since solutions for a and b exist

624
00:37:26.828 --> 00:37:29.649
this means a solution exists

625
00:37:29.649 --> 00:37:32.309
In other words

626
00:37:32.309 --> 00:37:36.172
combining (2,1) and (1,3)

627
00:37:36.172 --> 00:37:42.800
we can generate all vectors on the plane

628
00:37:42.800 --> 00:37:46.704
If we pick any two vectors

629
00:37:46.704 --> 00:37:51.520
can we always generate all vectors on the plane?

630
00:37:51.520 --> 00:37:55.555
Let's consider (1,2) and (2,4)

631
00:37:55.555 --> 00:38:00.570
Representing an arbitrary vector (x,y) with these

632
00:38:00.570 --> 00:38:03.973
we can rewrite and solve it as a system of equations

633
00:38:03.973 --> 00:38:10.080
If we double one equation, 2x - y = 0

634
00:38:10.080 --> 00:38:16.960
This implies 2x = y

635
00:38:16.960 --> 00:38:18.247
2x =y

636
00:38:18.247 --> 00:38:22.883
This means the solution exists only when

637
00:38:22.883 --> 00:38:26.760
x is twice y

638
00:38:26.760 --> 00:38:28.825
This implies solutions do not exist for all vectors

639
00:38:28.825 --> 00:38:35.325
but only when x and y

640
00:38:35.325 --> 00:38:37.820
satisfy specific conditions

641
00:38:37.820 --> 00:38:40.981
Representing this graphically

642
00:38:40.981 --> 00:38:46.552
we can only generate vectors of the form (x,2x)

643
00:38:46.552 --> 00:38:50.320
This is the only set of vectors we can create

644
00:38:50.320 --> 00:38:54.014
Visualized, it looks like this

645
00:38:54.014 --> 00:38:58.439
The vectors generated by (1,2) and (2,4)

646
00:38:58.439 --> 00:39:04.620
exist on the line y = 2x

647
00:39:04.620 --> 00:39:11.139
Only one-dimensional vectors on this line can be created

648
00:39:11.139 --> 00:39:14.935
This is similar to what we discussed earlier

649
00:39:14.935 --> 00:39:17.959
scalar multiplication with vectors

650
00:39:17.959 --> 00:39:19.004
Yes, that’s correct

651
00:39:19.004 --> 00:39:20.053
It’s the same concept

652
00:39:20.053 --> 00:39:22.760
Because (2,4)

653
00:39:22.760 --> 00:39:28.959
is simply (1,2) scaled by 2

654
00:39:28.959 --> 00:39:32.016
Thus, the equation

655
00:39:32.016 --> 00:39:37.991
can be rewritten as a(1,2) + 2b

656
00:39:37.991 --> 00:39:40.498
Since (1,2) is the same

657
00:39:40.498 --> 00:39:44.507
the distributive property combines them into (a + 2b)(1,2)

658
00:39:44.507 --> 00:39:49.387
This is equivalent to scaling (1,2) by a scalar

659
00:39:49.387 --> 00:39:52.159
Thus, it results in scalar multiplication of a vector

660
00:39:52.159 --> 00:40:00.919
which limits us to vectors on a one-dimensional line

661
00:40:00.919 --> 00:40:06.113
Therefore, some vector combinations

662
00:40:06.113 --> 00:40:10.191
only generate vectors on a one-dimensional line

663
00:40:10.191 --> 00:40:14.960
while others can generate all vectors on a plane

664
00:40:14.960 --> 00:40:17.397
These cases arise depending on the vectors

665
00:40:17.397 --> 00:40:22.454
So, when can we generate all vectors

666
00:40:22.454 --> 00:40:28.813
and when are we limited to one-dimensional vectors on a line?

667
00:40:28.813 --> 00:40:35.045
This depends on whether the vectors

668
00:40:35.045 --> 00:40:39.800
are linearly independent

669
00:40:39.800 --> 00:40:41.832
In conclusion

670
00:40:41.832 --> 00:40:45.844
this is possible only if they are linearly independent

671
00:40:45.844 --> 00:40:54.199
If solutions for a and b exist

672
00:40:54.199 --> 00:41:00.641
and satisfy the system of equations

673
00:41:00.641 --> 00:41:02.091
we can generate all vectors

674
00:41:02.091 --> 00:41:05.405
If no solutions exist, only specific vectors

675
00:41:05.405 --> 00:41:07.520
on a line can be generated

676
00:41:07.520 --> 00:41:10.918
This is the conclusion we can draw

677
00:41:10.918 --> 00:41:15.919
As for the conditions for the existence of solutions

678
00:41:15.919 --> 00:41:17.877
we’ll cover that in the next session

679
00:41:20.080 --> 00:41:23.971
To summarize

680
00:41:23.971 --> 00:41:31.713
to generate all vectors on a plane

681
00:41:31.713 --> 00:41:36.228
we need a combination of vectors

682
00:41:36.228 --> 00:41:38.095
that consists of two vectors

683
00:41:38.095 --> 00:41:42.011
and these vectors must be linearly independent

684
00:41:42.011 --> 00:41:45.716
We can sum it up in this way

685
00:41:45.716 --> 00:41:48.735
Before we explore this further

686
00:41:48.735 --> 00:41:51.937
and before delving into why

687
00:41:51.937 --> 00:41:54.381
let’s first examine

688
00:41:54.381 --> 00:41:57.002
the mathematical concepts of basis and dimension

689
00:41:57.002 --> 00:41:59.639
I’ll provide a brief explanation

690
00:41:59.639 --> 00:42:03.183
The definition of a basis

691
00:42:03.183 --> 00:42:07.560
is a set of linearly independent vectors

692
00:42:07.560 --> 00:42:13.726
that can generate all vectors in a vector space

693
00:42:13.726 --> 00:42:17.397
A set of vectors exists

694
00:42:17.397 --> 00:42:19.140
All the elements are vectors

695
00:42:19.140 --> 00:42:23.654
But these vectors must be linearly independent

696
00:42:23.654 --> 00:42:26.234
This defines a basis

697
00:42:26.234 --> 00:42:31.226
These independent vectors, through linear combinations

698
00:42:31.226 --> 00:42:35.743
can generate all vectors in the vector space

699
00:42:35.743 --> 00:42:39.276
These vectors are called

700
00:42:39.276 --> 00:42:43.560
the basis of the vector space

701
00:42:43.560 --> 00:42:47.110
The basis is a set-based concept

702
00:42:47.110 --> 00:42:51.019
The elements of the basis are vectors

703
00:42:51.019 --> 00:42:54.858
These vectors, which are elements of the basis

704
00:42:54.858 --> 00:42:56.531
are called basis vectors

705
00:42:56.531 --> 00:42:59.870
We refer to them as basis vectors in English

706
00:42:59.870 --> 00:43:04.708
Another important concept is dimension

707
00:43:04.708 --> 00:43:07.000
When we talk about dimensions

708
00:43:07.000 --> 00:43:09.993
we often think of 1D, 2D, 3D, 4D, and so on

709
00:43:09.993 --> 00:43:12.377
These correspond to how we perceive space

710
00:43:12.377 --> 00:43:14.908
3D for volume, 2D for a plane

711
00:43:14.908 --> 00:43:17.639
as we commonly describe them

712
00:43:17.639 --> 00:43:20.247
However, mathematically, dimensions

713
00:43:20.247 --> 00:43:24.446
are not about how we perceive space

714
00:43:24.446 --> 00:43:27.941
Instead, they refer to the number of basis vectors

715
00:43:27.941 --> 00:43:30.399
needed to form the space

716
00:43:30.399 --> 00:43:33.022
It’s the number of elements in the basis

717
00:43:33.022 --> 00:43:38.017
For example, a 2D space means

718
00:43:38.017 --> 00:43:42.595
there exists a space where all vectors

719
00:43:42.595 --> 00:43:47.252
can be generated as linear combinations

720
00:43:47.252 --> 00:43:48.431
of a set of basis vectors

721
00:43:48.431 --> 00:43:53.449
The number of elements in the basis is the dimension

722
00:43:53.449 --> 00:43:55.024
Thus, 2D means

723
00:43:55.024 --> 00:43:58.463
there are exactly two basis vectors

724
00:43:58.463 --> 00:44:01.620
This is what defines 2D

725
00:44:01.620 --> 00:44:04.999
If I just explain the definitions

726
00:44:04.999 --> 00:44:07.450
it might feel a bit abstract

727
00:44:07.450 --> 00:44:10.159
So, I’ve prepared an example

728
00:44:10.159 --> 00:44:13.355
Since we’re discussing a 2D plane

729
00:44:13.355 --> 00:44:17.919
this means the basis always has two vectors

730
00:44:17.919 --> 00:44:21.617
Previously, to generate (5,5)

731
00:44:21.617 --> 00:44:23.693
we used two combinations of vectors

732
00:44:23.693 --> 00:44:26.293
(1,0) and (0,1)

733
00:44:26.293 --> 00:44:28.899
and (2,1) and (1,3)

734
00:44:28.899 --> 00:44:33.032
These are called basis sets or simply bases

735
00:44:33.032 --> 00:44:35.798
The term "basis" is a shorthand for basis sets

736
00:44:35.798 --> 00:44:38.422
Let’s denote a basis as b1, and b2

737
00:44:38.422 --> 00:44:41.307
Beyond these, as previously discussed

738
00:44:41.307 --> 00:44:44.207
any solution to the system of equations forms a basis

739
00:44:44.207 --> 00:44:47.130
This is because they are linearly independent

740
00:44:47.130 --> 00:44:51.399
Thus, any pair of vectors that solves the system

741
00:44:51.399 --> 00:44:54.399
must always consist of exactly two vectors

742
00:44:54.399 --> 00:44:57.419
Not three, not one, not four

743
00:44:57.419 --> 00:44:59.329
There are only two

744
00:44:59.329 --> 00:45:01.170
Linearly independent vectors

745
00:45:01.170 --> 00:45:03.744
form the basis elements for a plane

746
00:45:03.744 --> 00:45:07.239
and there are exactly two of them

747
00:45:07.239 --> 00:45:10.989
While there are infinitely many possible bases

748
00:45:10.989 --> 00:45:14.998
the number of elements in each basis, its dimension

749
00:45:14.998 --> 00:45:18.820
is always two

750
00:45:18.820 --> 00:45:20.713
If there is only one basis vector

751
00:45:20.713 --> 00:45:23.135
as explained earlier, what would happen?

752
00:45:23.135 --> 00:45:27.151
Multiplying a single basis vector by scalars

753
00:45:27.151 --> 00:45:31.791
produces vectors along a specific slope

754
00:45:31.791 --> 00:45:35.726
that lie on a single line in 1D

755
00:45:35.726 --> 00:45:37.644
This generates a 1 dimension space

756
00:45:37.644 --> 00:45:43.140
not all vectors in a 2 dimension plane

757
00:45:43.140 --> 00:45:47.923
Thus, only vectors on a single line can be generated

758
00:45:47.923 --> 00:45:53.379
If there are more than three vectors

759
00:45:53.379 --> 00:45:55.188
What this means is

760
00:45:55.188 --> 00:45:58.592
using two linearly independent vectors

761
00:45:58.592 --> 00:46:02.820
we can generate all points, as mentioned earlier

762
00:46:02.820 --> 00:46:08.367
If we claim three vectors are linearly independent

763
00:46:08.367 --> 00:46:11.104
then the first two vectors

764
00:46:11.104 --> 00:46:15.300
v1 and v2 are independent

765
00:46:15.300 --> 00:46:19.563
But any vector scaled by a3

766
00:46:19.563 --> 00:46:22.003
can already be generated

767
00:46:22.003 --> 00:46:25.115
In a 2D plane

768
00:46:25.115 --> 00:46:28.459
these two vectors

769
00:46:28.459 --> 00:46:34.739
can already generate all others via linear combinations

770
00:46:34.739 --> 00:46:39.819
So, even if a3 is not 0, what happens?

771
00:46:39.819 --> 00:46:43.018
it results in the zero vector

772
00:46:43.018 --> 00:46:46.599
This violates the definition of linear independence

773
00:46:46.599 --> 00:46:48.280
explained earlier

774
00:46:48.280 --> 00:46:52.089
Thus, the three vectors are linearly dependent

775
00:46:52.089 --> 00:46:55.635
and they fail to satisfy the condition

776
00:46:55.635 --> 00:46:57.817
of a set of linearly independent vectors

777
00:46:57.817 --> 00:47:02.300
Therefore, they cannot form a basis

778
00:47:02.300 --> 00:47:07.061
When defining dimensions

779
00:47:07.061 --> 00:47:10.491
we count the number of linearly independent vectors

780
00:47:10.491 --> 00:47:14.020
that span the space

781
00:47:14.020 --> 00:47:17.277
This is used to define

782
00:47:17.277 --> 00:47:19.409
the vector space’s dimension

783
00:47:19.409 --> 00:47:21.372
For example, R2, R3

784
00:47:21.372 --> 00:47:22.850
As mentioned earlier

785
00:47:22.850 --> 00:47:24.543
in vector spaces

786
00:47:24.543 --> 00:47:27.203
we use real numbers

787
00:47:27.203 --> 00:47:30.216
and thus real vector spaces

788
00:47:30.216 --> 00:47:34.399
So, how many dimensions are formed by Cartesian products?

789
00:47:34.399 --> 00:47:39.737
For 2D, we combine the x and y axes

790
00:47:39.737 --> 00:47:41.930
orthogonally intersecting them

791
00:47:41.930 --> 00:47:45.844
For 3D, we add a third axis perpendicular to both

792
00:47:45.844 --> 00:47:48.699
forming three intersecting axes

793
00:47:48.699 --> 00:47:51.088
This creates a 3D space

794
00:47:51.088 --> 00:47:54.634
These are how we construct vector spaces

795
00:47:54.634 --> 00:47:57.330
Since we cannot percieve 4-dimensions

796
00:47:57.330 --> 00:47:59.776
if we add an unseen perpendicular axis

797
00:47:59.776 --> 00:48:02.498
to form a fourth dimension

798
00:48:02.498 --> 00:48:05.540
a 4D space can be constructed theoretically

799
00:48:05.540 --> 00:48:08.080
Beyond our perception

800
00:48:08.080 --> 00:48:11.915
we can continue adding orthogonal axes

801
00:48:11.915 --> 00:48:14.659
to construct n-dimensional spaces

802
00:48:14.659 --> 00:48:17.490
Mathematically, we can construct an n-dimensional space

803
00:48:17.490 --> 00:48:22.436
The, in such an n-dimensional space

804
00:48:22.436 --> 00:48:26.540
the number of linearly independent vectors that span all vectors

805
00:48:26.540 --> 00:48:29.629
is n, and this is denoted

806
00:48:29.629 --> 00:48:32.584
by adding a subscript to form Rn

807
00:48:32.584 --> 00:48:36.500
This is how we define a vector space

808
00:48:36.500 --> 00:48:39.867
Looking back at the diagram

809
00:48:39.867 --> 00:48:41.782
a basis is a set-based concept

810
00:48:41.782 --> 00:48:44.044
and the elements of the basis

811
00:48:44.044 --> 00:48:46.609
are called basis vectors

812
00:48:46.609 --> 00:48:50.080
The dimension refers to the number of elements

813
00:48:50.080 --> 00:48:53.640
in the basis set

814
00:48:53.640 --> 00:49:00.038
The simplest basis to work with

815
00:49:00.038 --> 00:49:04.179
is the one we first discussed, which was (1,0), and (0,1)

816
00:49:04.179 --> 00:49:06.383
These vectors have a magnitude of 1

817
00:49:06.383 --> 00:49:09.953
and correspond exactly to

818
00:49:09.953 --> 00:49:14.436
one element in each coordinate axis

819
00:49:14.436 --> 00:49:17.766
They are used precisely and exclusively

820
00:49:17.766 --> 00:49:19.906
as the standard for constructing space

821
00:49:19.906 --> 00:49:25.787
These axes evenly form the three dimensions

822
00:49:25.787 --> 00:49:27.979
and serve as the standard for space construction

823
00:49:27.979 --> 00:49:30.481
These basis vectors are called

824
00:49:30.481 --> 00:49:33.340
standard basis vectors

825
00:49:33.340 --> 00:49:35.930
Standard basis vectors

826
00:49:35.930 --> 00:49:40.875
are denoted using special symbols such as e1, e2

827
00:49:40.875 --> 00:49:42.163
In the vector space R2

828
00:49:42.163 --> 00:49:47.199
the standard basis vectors are e1 = (1,0)

829
00:49:47.199 --> 00:49:49.219
and e2= (0,1)

830
00:49:49.219 --> 00:49:53.486
In R3, the standard basis vectors are e1 = (1,0,0)

831
00:49:53.486 --> 00:49:55.960
e2= (0,1,0)

832
00:49:55.960 --> 00:49:59.179
and e3=(0,0,1)

833
00:49:59.179 --> 00:50:05.227
This covers the key concepts of

834
00:50:05.227 --> 00:50:08.798
basis and dimension that form spaces

835
00:50:08.798 --> 00:50:11.402
To summarize

836
00:50:11.402 --> 00:50:16.131
a basis is the foundation

837
00:50:16.131 --> 00:50:18.659
or cornerstone of constructing a space

838
00:50:18.659 --> 00:50:23.889
The most straightforward and common form 
is the standard basis vectors

839
00:50:23.889 --> 00:50:27.766
However, any set of linearly independent vectors

840
00:50:27.766 --> 00:50:31.060
can serve as a basis

841
00:50:31.060 --> 00:50:34.777
as we discussed

842
00:50:34.777 --> 00:50:37.820
We can summerize it like this

843
00:50:37.820 --> 00:50:41.054
As for dimension, let me emphasize again

844
00:50:41.054 --> 00:50:47.089
it is not about how we perceive or feel space

845
00:50:47.089 --> 00:50:52.924
In mathematics, dimension refers to 
the number of linearly independent basis vectors

846
00:50:52.924 --> 00:50:55.908
the number of elements in the basis

847
00:50:55.908 --> 00:50:57.699
We can sum it up like this

848
00:50:57.699 --> 00:50:59.941
This concludes today’s lecture

849
00:50:59.941 --> 00:51:01.884
Thank you for listening

850
00:51:01.884 --> 00:51:02.905
Thank you

851
00:51:03.238 --> 00:51:05.580
Vector
Definition of vector
There is a limitation in that it is expressed only as a point on a straight line
Cartesian coordinate system: A method of expressing a plane by creating a new line that intersects a straight line at 90 degrees

852
00:51:05.580 --> 00:51:07.880
Multiplication of vectors and scalars: Creates a new vector on a line of gradient that includes the + and - directions based on the origin
Size of vector: The shortest distance from the origin

853
00:51:07.880 --> 00:51:10.218
Vector-scalar multiply: Creates a new vector along a slope covering pos and neg directions
Vector magnitude: Shortest distance from the origin

854
00:51:10.223 --> 00:51:13.723
Vector
Axioms of Vector Spaces, Associativity of addition, Commutativity of addition,
Additive inverse, Scalar multiplication, Compatibility of operations

855
00:51:13.723 --> 00:51:17.243
Identity of scalar multiplication operation
Distributive property of vector addition and scalar multiplication
Distributive property of scalar addition and scalar multiplication

856
00:51:17.243 --> 00:51:18.560
Linear Independence
Vector Generation System
You can generate a vector on a one-dimensional line or plane depending on the vector combination
Linear Combination: A formula that generates a new vector using the basic operations of vectors

857
00:51:18.560 --> 00:51:19.710
Span: A mathematical expression for generating a new vector
Linear Dependence: When there is a coefficient whose value of a_n is not 0 in the equation of a linear combination that generates a 0 vector

858
00:51:19.710 --> 00:51:20.908
Linear independence:When any coefficient in linear combination for the 0 vector is 0
Only linear independent vector can generate all vectors in plane

859
00:51:20.908 --> 00:51:22.085
Basis and Dimension  
Basis: A set of linearly independent vectors that can generate all vectors in a vector space
Basis vector: An element of basis set

860
00:51:22.085 --> 00:51:23.209
Standard basis vector: The most fundamental vector 
Dimension: number of elements in the basis set
2D: Formed by x, y axes
3D: formed by x, y, z, axes