Matrix Transformation

Question 1

What’s domain, co-domain and range of a function?

Answer 1

Domain: space from which the input is drawn
Co-domain: space to which an input is transformed by the function
Range: Set of function outputs

Question 2

What’s a vector valued function?

A more formal understanding of functions 15:15

Answer 2

A vector valued function is a function where the range is a vector (it maps the values to R² or higher space co-domains)

Question 3

What is another name for vector functions?

Vector Transformation 11:10

Answer 3

Transformation

Question 4

What’s a linear transformation?

Linear Transformation 00:00

Answer 4

A function is a linear transformation if and only if:
It’s a transformation that maps from Rⁿ to R^m
And for a,b in Rⁿ it satisfies two conditions:
1) T(a+b)=T(a)+T(b)
2) T(ca)=cT(a)
* c is scalar

Question 5

How can we find the new coordination of a point, after applying a linear transformation on R²?

Matrix from visual representation of transformation

Answer 5

We should see where [1 0] and [0 1] land after transformation, then use linear transformation rules (T(a+b)=T(a)+T(b) and T(ca)=cT(a)) to find where a given point lands

Question 6

Matrix products with a vector is always a ____

Matrix vector product as linear transformation 14:50

Answer 6

Linear transformation

Question 7

All linear Transformations can be a matrix-vector product. True/False. Why?

Linear transformation as matrix vector product 10:54

Answer 7

True, because all we need to do is to calculate the result of the transformation for standard basis of the domain ([1 0] [0 1] for 2D), then multiply that, by whatever vector we have (input of transformation). This equals the vector’s transformation, therefore each linear transformation can be written as a matrix-vector product 11:10

Question 8

We know that we can define a line in 2D or a higher space like this: ____
and we know that every matrix transformation is a ____ transformation
How can we write down the formula for matrix transformation of line L: T(L)?

Image of a subset under a transformation

Answer 8

{A+t(B-A)|t ∈ R}, A and B are vectors (vectors start from origin and end at a point’s coordination, so these two are vectors corresponding to two points on L)

Linear transformation

T(L)={T(A)+t (T(A)-T(B))|t ∈ R}
This means: in order to find T(L) we need to know the coordinates of two points on it

Question 9

T(L) is called the ____ of L under the transformation T

Answer 9

Image

Image of a subset under a transformation 16:30

Question 10

What is the image of a transformation?

im(T): Image of a transformation 10:47

Answer 10

When we transform a whole n dimensional space, it’s called Image of T.

technically it’s called image of Rⁿ under T
T( Rⁿ)={T(x)|x∈Rⁿ}

Question 11

Projection is a linear transformation. True/False

External

Answer 11

True

Question 12

Is T(x)=T(I)x?why?
I: Identity matrix
x: column vector

Answer 12

Yes.
T(x)=T(Ix)=T(V₁x₁+…+V_nx_n)= T(V₁x₁)+…+T(V_nx_n)=
x₁T(V₁)+…+x_nT(V_n)=
T([V₁ … V_n] )[x₁ … x_n]=T(I)x
.
[x₁ … x_n] is a column vector

Vs are the vectors(columns) of the identity matrix

Note: T(I) is usually denoted by “A”, aka. transformation matrix

Question 13

Is a composition of two linear transformations (applying two consecutive linear transformation on vector x), a linear transformation?

Composition of linear transformations pt1 07:11

Answer 13

Yes