Game Basics The Fundamental Mathematics of Games: Vectors and Linear Independence
Hello, everyone
This is Lee Deok-woo, from Game Mathematics
In this session, as explained in the previous session
we will delve into vectors
which form the foundation of game systems
I will cover the definition of vectors, operations involving vectors
and the axioms that constitute vector spaces
Then, we will explore linear independence and dimensions
The discussion is divided into two parts
First is understanding the generation system of vectors
within vector spaces
And next is exploring the mathematical concepts of basis and dimensions
Vectors
Let’s begin by explaining the definition of vectors
and understanding what vectors really are
In the previous session, we learned about numbers
Numbers are simply sets of elements
that can be visualized
as a line of points arranged seamlessly
Additionally, numbers are not just sets
based on axioms
they are governed by binary operations
allowing new numbers to be generated
However, when we consider the visual representation of numbers
they are limited to being
expressed as points on a straight line
When we want to represent something through a point on a straight line
this limitation becomes clear
To depict anything of visual significance
we need at least a plane
where we can draw
and manipulate various shapes
Using only numbers confined to a straight line
makes it challenging to express much
To address this, we need to extend numbers beyond one-dimensional space
If the domain of numbers is considered one-dimensional
we can say that we need to extend it to a two-dimensional domain
Let’s explore how this extension is achieved
In mathematics, the two-dimensional representation method commonly used
is where the x-axis points to the right and the y-axis points upward
and we usually call this a graph
This representation is widely known as
the Cartesian Coordinate System
In English, it is called the Cartesian Coordinate System
The Cartesian coordinate system is created
by taking a straight line, representing numbers
and intersecting it perpendicularly with another straight line
to form a plane
This perpendicular intersection forms
the basis of the Cartesian coordinate system
In this system, the positive direction
commonly called the plus direction
points to the right and upward
As shown in the diagram, the rightward and upward directions
are defined as independent sets
of real numbers
which intersect at 90 degrees
The perpendicular intersection is referred to as "orthogonality"
which carries a significant meaning
To briefly explain, orthogonality means
more than just a 90-degree arrangement
it signifies that the set of real numbers along the horizontal axis
and the set of real numbers along the vertical axis are unrelated
This is the essence of the 90-degree intersection
By creating this extended plane
we now have a stage
to work with
visually meaningful elements
From the perspective of numbers as sets
this plane can be seen as
the Cartesian product of two sets of real numbers, R x R
The elements of this Cartesian product
are expressed using a notation called tuples
which use parentheses and commas
if X represents an element of the first set of real numbers
and Y represents an element of the second set
represented as an element of the Cartesian product
as (X, Y)
This element is used to describe the Cartesian product visually
and we call this a coordinate
So we say that a coordinate is used to describe the Cartesian product
These coordinates are typically represented
when visualized
as points on the plane
where the x-axis and y-axis intersect perpendicularly
As we discussed earlier, numbers can be represented in two ways
as points at specific locations
or as arrows
emanating from the origin
Thus, an element of this two-dimensional space
the Cartesian product of two sets of real numbers
can be represented as a point or as an arrow
originating from the origin
The choice between these two representations
as a point or as an arrow
depends on the context
When we discuss affine spaces later, we’ll clarify when to use what
For now, it suffices to understand that
both representations are possible
Now, if we revisit the straight line
represented as a set of numbers
and the plane expressed through Cartesian products
from the perspective of sets and elements
In the case of a straight line, it is represented by a single element
from the set of real numbers
Now, with ordered pairs as elements of the Cartesian product
expressed as x, y, we can represent elements on the plane
This establishes the basis for representing elements
So, what do we call this set?
We can call it a Cartesian product
or refer to it as elements of the Cartesian product
However, describing it as a Cartesian product of real numbers
or elements of a Cartesian product of real numbers
is quite long and difficult
This is because
when we use computer graphics in the future
it becomes the foundation for everything
Such complex explanations are inconvenient
That is why we decided to give these sets and elements
distinct names to use and call them by
Those names are vectors and vector spaces
As shown in the diagram
we define vectors and vector spaces separately
In the two-dimensional plane
and extending further to three dimensions and beyond
the concepts of numbers and sets of numbers
are generalized into vectors and vector spaces
This can be viewed as an extension
As I mentioned at the end of Lecture 1
analyzing and representing the structure of number sets
from a broader perspective is more versatile
It allows for broader applications
Therefore, vector and vector space refer to
number sets
which in this context use real numbers
but from the perspective of fields
they form a Cartesian product
Understanding this from a broader viewpoint is helpful
Thus, vector spaces
can be seen as Cartesian products of two fields
from the perspective of sets
The elements of these fields are called scalars
Combining these scalars into ordered pairs
defines a vector
This structural perspective
enables the definition of vector spaces and vectors
The vector space is a concept of sets
and vectors are its elements
This provides a systematic framework
If we think of them as simple sets
vectors merely represent static objects
They are just elements within sets
However, to represent something dynamic over time
like forming a system that creates animations
the system of vector spaces
requires binary operations similar to number sets
These are called the fundamental operations of vectors
Regarding the fundamental operations of vectors
I’ve summarized two main types
The first is vector addition
The concept of a vector
initially begins as the Cartesian product of two sets
but adding another field results in
a Cartesian product of three, four, and so on
This can be extended indefinitely
To simplify
let’s consider the Cartesian product of two sets
Here, an element can be expressed as the ordered pair a, b
If another element is c, d
vector addition is defined as follows
Add the corresponding components of each pair
This independent addition forms vector addition
When this vector addition is visualized
it appears like this
For example, adding (1,2) and (3,1)
Add the x-values 1 + 3
and add the y-values 2 + 1
This gives the vector addition result
How can I explain this?
If I compare this mechanism
to real-world substances
It’s like water and oil
Since x and y are perpendicular
we said they are unrelated, right?
They cannot mix
They cannot influence each other
A large x-value does not affect y
and no matter how big or small y becomes
it does not influence x
These are two independent sets
This can be compared to water and oil
Now, let’s consider (1,2)
as 10 ml of water and 20 ml of oil
and (3, 2) as 30 ml of water and 10 ml of oil
If you combine them in a cup, what would happen?
Obviously, water and oil would remain separate
Thus, vector addition
is like water and oil along the x-axis
add x-values and y-values separately
as they do not influence each other
and this completes the calculation
This is the first fundamental operation of vectors
You might think that the next one is multiplication between vectors
Your guess is wrong
It is the multiplication of a vector and a scalar
As explained earlier, a scalar refers to
an element of a field set
We call the elements of a field set scalars, right?
If we use the set of real numbers as our field
then scalars here are real numbers
Thus, it could be described as multiplication between vectors and real numbers
But calling it the multiplication of vectors and scalars
provides a broader and more structural perspective
It offers a clearer understanding of the concept
So, the multiplication of a vector and a scalar
is represented as k, where k is a scalar value
When multiplying a vector (a, b) by k
you get ka, kb, and reversing the order gives the same result
Reversing it would give ak, bk
Here, a and b are both scalars
and the multiplier k is also a scalar
so ka, kb, and ak, bk
may be different or the same
However, based on the axioms of fields
we know that multiplication is commutative
Therefore, ka equals ak
as defined by the axioms of fields
Therefore, they produce the same result
In conclusion, multiplying (a,b) by k has the same result as multiplying each component of a,b by k
This is verified by the axioms of fields
This multiplication of vectors and scalars
is adopted as the second fundamental operation
When graphing this multiplication
the result is as follows
When we represent a vector as an arrow originating from the origin
imagine it visually
The arrow from the origin has
what we commonly call a slope
This slope defines a line that perfectly aligns with the arrow
It can be seen as a line
This is essentially an infinite line
The line follows the slope of the vector
On this infinite line
any new vector created by scalar multiplication
will always lie on the infinite line
It will always be on this green line
Thus, scalar multiplication of vectors
can be seen as a method for creating one-dimensional vectors
This is because they exist only along the line
To summarize the two concepts
First, vector addition
moves x-values and y-values
independently of each other
resulting in a parallel shift by x and y
This produces a visual representation of parallel movement
Second, scalar multiplication of a vector
creates a vector centered on the origin
The scalar multiplication of that vector
lies on the infinite line with the same slope
It exists only along that invisible line
This is how a vector is generated
Previously, when we discussed numbers
we understood positive and negative directions originating from the origin
I explained that understanding it this way helps
This is the same concept
Usuing (0,0) as the reference point
it encompasses both positive and negative directions
On a line with a specific slope
a system generates new vectors
This can be seen as a system for creating one-dimensional vectors
These are the two fundamental operations
In practical applications
these two operations are not commonly used directly
You’ve likely heard of them
dot products and cross products of vectors
These are widely used in practical applications
They are operations meant specifically for applications
We will learn about them later
At this stage, as we learn the definition of vectors
we focus on these two fundamental operations
Consider this as the focus of our current learning
Another important concept in vectors is their magnitude
Previously, we discussed the magnitude of numbers
as the shortest distance from the origin
The same applies to vectors
The shortest distance from the origin
is defined as the magnitude of a vector
This can be referred to as the distance from the origin
the length from the origin
the vector’s length
or in mathematical terms, the norm
These three terms essentially mean the same
To calculate the magnitude or norm
it also represents the shortest distance from the origin
In a plane, the shortest distance from the origin
is, as shown in the diagram
a right triangle
It is the distance of the hypotenuse of the right triangle
So, how can we calculate this distance?
You are probably familiar with the Pythagorean theorem
It states that the square of the base plus the square of the height
equals the square of the hypotenuse
this is expressed as a^2 + b^2 = c^2
Using this
we can calculate the shortest distance from the origin
This distance is
the square root of x^2 + y^2
Thus, this shortest distance, as we discussed earlier
is similar to finding the shortest distance for a number
by using absolute value symbols
We denote this by placing vertical bars around the number
By using absolute value symbols
we can find the shortest distance
Similarly, for vectors
if we place vertical bars around a vector
it represents the shortest distance from the origin
or the magnitude of the vector
Generally, using double vertical bars
is more commonly used
In this course, however
to simplify the expressions
we will use only a single vertical bar
Using double bars can make the notation cumbersome
To ensure consistency in meaning
we will use a single vertical bar to denote magnitude
Like this
Once we have determined the magnitude of a vector
the next concept to understand is
the unit vector
A unit vector is a vector with a magnitude of 1
To find a unit vector
consider the original magnitude of the vector
By multiplying this magnitude by its reciprocal
the multiplicative inverse
the magnitude becomes 1
This results in the multiplicative identity
When you multiply a number by its inverse
you get the identity element 1
The same principle applies here
That covers the explanation of vector magnitude
Finally, based on these fundamental elements
let’s explore how the vector system
is structured
As mentioned briefly earlier
a vector, from the perspective of sets
can be seen as a collection of points
However, if we view it as a closed system
a closed system able to generate
new elements
then we need to organize the operations
The two operations mentioned earlier—vector addition
and vector-scalar multiplication
must exist to enable the creation
of new vectors within the system
These operations aren’t just arbitrarily defined
As with number sets
we organize axioms for operations within a field
If these axioms are satisfied
we can define a system accordingly
This allows us to structurally
view the vector space system
It provides the foundation for this perspective
Thus, a vector space
is a system encompassing vectors
The broad concept of a vector space
is also governed
by several axioms
It is composed of 8 axioms
Let’s examine these axioms
Here, u, v, and w represent vectors
and a and b represent scalars
The first axiom is the associative property of addition
Addition refers to vector addition
The associative property means that
the grouping of additions doesn’t affect the result
For example, with vectors u, v, and w
adding v and w first, and then u
or adding u and v first, then w
produces the same result
This is because each component of a vector is a scalar
The x-components are added separately from the y-components
Since x and y are elements of a field
the associative property naturally holds
Thus, even for vectors
this property holds true
The first axiom of associativity is satisfied
The second is the commutative property of addition
This, too, is inherent in vector systems based on fields
Naturally, it must hold true
The third is the existence of an additive identity
This refers to the zero vector, where all components are zero
Adding 0 to each component leaves the vector unchanged
so the vector's value remains the same
Therefore, in vector addition
the additive identity in a vector space
is defined as the zero vector
From here, we can define the additive inverse
The additive inverse of a vector is
the vector obtained by negating it
This represents a vector pointing in the opposite direction
so the additive inverse is
denoted as -v
Now, let’s move on to multiplication
Multiplication here does not involve vectors multiplying each other
Instead, it is the multiplication of a vector by a scalar, which might feel a bit different
This is because vectors
and vector spaces are not solely composed of vectors
As mentioned earlier, the foundation
that forms a vector space lies
in the structure of a field set
Thus, vector spaces operate over a field
and are built on this field
As a result, scalars
must always be considered
This is a fundamental difference
between number sets and vectors
Scalar multiplication involves compatibility
a property known as compatibility in English
No matter which scalar is used
changing the order does not alter the result
This is because it satisfies the commutative and associative properties of multiplication
Hence, it naturally holds true
When multiplying a scalar by 1
each component is multiplied by 1
This is due to the multiplicative identity
so it results in the same vector
Thus, the identity for scalar multiplication
is defined as 1
Next, when both addition and multiplication
are used together
we previously defined this as the distributive property
When working with vectors
we deal with both vectors and scalars
If the elements in addition are vectors
it results in a form like a(u+v)
and this is equivalent to multiplying each vector by the scalar individually
Thus, the distributive property holds
If the elements in addition
are scalars instead of vectors
the result is still the same
It equals the sum of the products of the scalars
This indicates that the distributive property has two variations
With this, we have covered all 8 axioms
that define the system of vector spaces
Finally, the definition of a vector space
is a system that extends the elements of a field set
by combining them into ordered pairs
This is the expanded concept
When using this practically
we need to specify the number set
The most commonly used set is
the set of real numbers
Thus, when using vector spaces, we often use the set of real numbers
To build a vector space
requires specific actions in applications
A system based on real numbers
is called
a real vector space
This serves as a practical stage for various applications
From now on, in the broader context of vector spaces
we will use
real vector spaces
to represent objects on a plane
This summarizes the direction of future discussions
Linear Independence
In this session, we will explore the concepts of linear independence and dimension
Today's discussion consists of two parts
First, we will examine the system for generating vectors
within a vector space
Next, we will explore the mathematical concepts
of basis and dimension
Let’s start with the vector generation system
A vector generation system
uses the fundamental operations
of vectors discussed earlier
vector addition and vector-scalar multiplication
By applying these two operations
we create new vectors in the system
This is similar to the number system discussed in the first session
where numbers are combined through binary operations
to generate new numbers
It works in a similar manner
This generation system
is called a linear combination
In mathematics, creating new vectors this way
is referred to as span
It does not use the term 'create'
but instead uses 'span'
Let’s explore how spanning works
and examine the relevant equations
As shown here, by multiplying a scalar a with a vector v
and summing these scalar-vector products
you can combine the elements
The result is
a vector because a scalar times a vector is still a vector
Adding vectors also results in a vector
Thus, a new vector is generated
If the resulting vector is denoted as v'
we can define the general formula for generating vectors like this
So, this n
allows us to sun as many terms as needed
By adding as much as necessary
we can generate new vectors
You can think of it this way
Now, there are two important concepts to understand here
They are linear dependence and linear independence
So, when two vectors
or n vectors are dependent on one another
or independent of each other
we say they are independent
These are the two key concepts
Rather than vaguely saying they overlap
or that they are separate
we shouldn't explain it like this
but we must define dependence and independence mathematically
clearly using equations
In mathematics, linear dependence is defined as follows
If a linear combination of vectors satisfies this equation
and results in the zero vector
then we have linear dependence
In any linear combination that produces the zero vector
if any of the coefficients a1, a2, a3, ..., an
are non-zero
it is defined as linear dependence
This is the formal definition of linear dependence
If all a's are non-zero
it is a case of linear dependence
The definition of linear independence is similar but different
If a linear combination produces the zero vector
but at least one coefficient is zero
not all coefficients need to be non-zero
The vectors in such an equation are linearly independent
This implies that is even one is 0, the linear independence does not hold
All coefficients must not be zero
If any coefficient is 0
it breaks the condition of linear independence
This is how it works
Now, you might wonder what these terms really mean
It may not feel intuitive yet
Let’s look at two simple examples
to examine linear combinations
and understand the relationship between dependence and independence
We’ll explore this in detail
I’ve prepared two problems for us
Are the vectors (1,1) and (2,2) linearly dependent or independent?
Here is the question
Let’s assign coefficients a1
to (1,1) and a2 to (2,2)
The coefficients a1 and a2 are scalars
If these coefficients produce the zero vector
and if a1 and a2 are non-zero
this indicates linear dependence
For example, if a1 = 2
and a2 = -2, the result is the zero vector
Thus, we say these vectors
are linearly dependent
Now, let’s consider (1,2)
in a different example
Are (1,2) and (2,1) dependent or independent?
To solve this problem
we’ll use scalar multiplication with vectors
and apply the axioms of vector spaces
We can expand the equations
After proceeding step by step
the x-component becomes a1 + 2a2
and the y-component becomes 2a1 + a2
If both components equal zero
we check if a solution exists
If a1 and a2 satisfy both equations
this determines their relationship
The only solution is
a1=0 and a2=0
Thus, the vectors (1,2) and (2,1)
are linearly independent
Now that we’ve defined linear dependence and independence
you might wonder why this is important
It is related to the system for generating new vectors
Let’s assume we want to create a vector using linear combinations
Suppose we want to generate the vector (5,5)
We’ll explore how to do this using linear combinations
Let’s assume a1, and a2
are the only coefficients we use
If we combine a1v1 and a2v2 to create the vector
I chose two vectors
(1,0) and (0,1)
Can we use them to linearly combine and form (5,5)?
If we multiply each by 5, as shown
we can generate (5,5) through linear combinations
This might seem too obvious
But are there other pairs that can achieve the same?
For instance, consider the vectors (1,3) and (2,1)
We can combine these two vectors
to create (5,5)
For example, if we mutiply (1,3) by 2
and (2,1) by 1
we can create (5,5) through a linear combination
So, it’s not just these two vectors
you could use other vectors as well
to generate (5,5), right?
You might ask this question
There are many linear combinations of other vectors
that can generate (5,5)
For example, by combining (1,2)
you can generate not just (5,5)
but any vector on the plane
How can we do this?
If we assume an arbitrary point (x,y) on the plane
although we do not know the coefficient
we define a1 and b1 as a and b
Then, the equation becomes
a system of linear equations
where 2a + b= x
and a + 3b = y
This forms a system of equations
Can we solve this system of equations?
Of course, we can
Multiply one equation by 2 and subtract the other
This gives -2b = x - 2y
Dividing by -2, we can find the solution for b
and once b is found
we can calculate a as well
Since solutions for a and b exist
this means a solution exists
In other words
combining (2,1) and (1,3)
we can generate all vectors on the plane
If we pick any two vectors
can we always generate all vectors on the plane?
Let's consider (1,2) and (2,4)
Representing an arbitrary vector (x,y) with these
we can rewrite and solve it as a system of equations
If we double one equation, 2x - y = 0
This implies 2x = y
2x =y
This means the solution exists only when
x is twice y
This implies solutions do not exist for all vectors
but only when x and y
satisfy specific conditions
Representing this graphically
we can only generate vectors of the form (x,2x)
This is the only set of vectors we can create
Visualized, it looks like this
The vectors generated by (1,2) and (2,4)
exist on the line y = 2x
Only one-dimensional vectors on this line can be created
This is similar to what we discussed earlier
scalar multiplication with vectors
Yes, that’s correct
It’s the same concept
Because (2,4)
is simply (1,2) scaled by 2
Thus, the equation
can be rewritten as a(1,2) + 2b
Since (1,2) is the same
the distributive property combines them into (a + 2b)(1,2)
This is equivalent to scaling (1,2) by a scalar
Thus, it results in scalar multiplication of a vector
which limits us to vectors on a one-dimensional line
Therefore, some vector combinations
only generate vectors on a one-dimensional line
while others can generate all vectors on a plane
These cases arise depending on the vectors
So, when can we generate all vectors
and when are we limited to one-dimensional vectors on a line?
This depends on whether the vectors
are linearly independent
In conclusion
this is possible only if they are linearly independent
If solutions for a and b exist
and satisfy the system of equations
we can generate all vectors
If no solutions exist, only specific vectors
on a line can be generated
This is the conclusion we can draw
As for the conditions for the existence of solutions
we’ll cover that in the next session
To summarize
to generate all vectors on a plane
we need a combination of vectors
that consists of two vectors
and these vectors must be linearly independent
We can sum it up in this way
Before we explore this further
and before delving into why
let’s first examine
the mathematical concepts of basis and dimension
I’ll provide a brief explanation
The definition of a basis
is a set of linearly independent vectors
that can generate all vectors in a vector space
A set of vectors exists
All the elements are vectors
But these vectors must be linearly independent
This defines a basis
These independent vectors, through linear combinations
can generate all vectors in the vector space
These vectors are called
the basis of the vector space
The basis is a set-based concept
The elements of the basis are vectors
These vectors, which are elements of the basis
are called basis vectors
We refer to them as basis vectors in English
Another important concept is dimension
When we talk about dimensions
we often think of 1D, 2D, 3D, 4D, and so on
These correspond to how we perceive space
3D for volume, 2D for a plane
as we commonly describe them
However, mathematically, dimensions
are not about how we perceive space
Instead, they refer to the number of basis vectors
needed to form the space
It’s the number of elements in the basis
For example, a 2D space means
there exists a space where all vectors
can be generated as linear combinations
of a set of basis vectors
The number of elements in the basis is the dimension
Thus, 2D means
there are exactly two basis vectors
This is what defines 2D
If I just explain the definitions
it might feel a bit abstract
So, I’ve prepared an example
Since we’re discussing a 2D plane
this means the basis always has two vectors
Previously, to generate (5,5)
we used two combinations of vectors
(1,0) and (0,1)
and (2,1) and (1,3)
These are called basis sets or simply bases
The term "basis" is a shorthand for basis sets
Let’s denote a basis as b1, and b2
Beyond these, as previously discussed
any solution to the system of equations forms a basis
This is because they are linearly independent
Thus, any pair of vectors that solves the system
must always consist of exactly two vectors
Not three, not one, not four
There are only two
Linearly independent vectors
form the basis elements for a plane
and there are exactly two of them
While there are infinitely many possible bases
the number of elements in each basis, its dimension
is always two
If there is only one basis vector
as explained earlier, what would happen?
Multiplying a single basis vector by scalars
produces vectors along a specific slope
that lie on a single line in 1D
This generates a 1 dimension space
not all vectors in a 2 dimension plane
Thus, only vectors on a single line can be generated
If there are more than three vectors
What this means is
using two linearly independent vectors
we can generate all points, as mentioned earlier
If we claim three vectors are linearly independent
then the first two vectors
v1 and v2 are independent
But any vector scaled by a3
can already be generated
In a 2D plane
these two vectors
can already generate all others via linear combinations
So, even if a3 is not 0, what happens?
it results in the zero vector
This violates the definition of linear independence
explained earlier
Thus, the three vectors are linearly dependent
and they fail to satisfy the condition
of a set of linearly independent vectors
Therefore, they cannot form a basis
When defining dimensions
we count the number of linearly independent vectors
that span the space
This is used to define
the vector space’s dimension
For example, R2, R3
As mentioned earlier
in vector spaces
we use real numbers
and thus real vector spaces
So, how many dimensions are formed by Cartesian products?
For 2D, we combine the x and y axes
orthogonally intersecting them
For 3D, we add a third axis perpendicular to both
forming three intersecting axes
This creates a 3D space
These are how we construct vector spaces
Since we cannot percieve 4-dimensions
if we add an unseen perpendicular axis
to form a fourth dimension
a 4D space can be constructed theoretically
Beyond our perception
we can continue adding orthogonal axes
to construct n-dimensional spaces
Mathematically, we can construct an n-dimensional space
The, in such an n-dimensional space
the number of linearly independent vectors that span all vectors
is n, and this is denoted
by adding a subscript to form Rn
This is how we define a vector space
Looking back at the diagram
a basis is a set-based concept
and the elements of the basis
are called basis vectors
The dimension refers to the number of elements
in the basis set
The simplest basis to work with
is the one we first discussed, which was (1,0), and (0,1)
These vectors have a magnitude of 1
and correspond exactly to
one element in each coordinate axis
They are used precisely and exclusively
as the standard for constructing space
These axes evenly form the three dimensions
and serve as the standard for space construction
These basis vectors are called
standard basis vectors
Standard basis vectors
are denoted using special symbols such as e1, e2
In the vector space R2
the standard basis vectors are e1 = (1,0)
and e2= (0,1)
In R3, the standard basis vectors are e1 = (1,0,0)
e2= (0,1,0)
and e3=(0,0,1)
This covers the key concepts of
basis and dimension that form spaces
To summarize
a basis is the foundation
or cornerstone of constructing a space
The most straightforward and common form is the standard basis vectors
However, any set of linearly independent vectors
can serve as a basis
as we discussed
We can summerize it like this
As for dimension, let me emphasize again
it is not about how we perceive or feel space
In mathematics, dimension refers to the number of linearly independent basis vectors
the number of elements in the basis
We can sum it up like this
This concludes today’s lecture
Thank you for listening
Thank you
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
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
Vector-scalar multiply: Creates a new vector along a slope covering pos and neg directions Vector magnitude: Shortest distance from the origin
Vector Axioms of Vector Spaces, Associativity of addition, Commutativity of addition, Additive inverse, Scalar multiplication, Compatibility of operations
Identity of scalar multiplication operation Distributive property of vector addition and scalar multiplication Distributive property of scalar addition and scalar multiplication
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
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
Linear independence:When any coefficient in linear combination for the 0 vector is 0 Only linear independent vector can generate all vectors in plane
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
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