Question

if A is a m*n matrix prove that t(x)=A(x) from R^n to R^m​

Answer 1

Answer:

Hello mate.

Step-by-step explanation:

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Linear Transformations

The two basic vector operations are addition and scaling. From this perspec-

tive, the nicest functions are those which “preserve” these operations:

Def: A linear transformation is a function T : R

n → R

m which satisfies:

(1) T(x + y) = T(x) + T(y) for all x, y ∈ R

n

(2) T(cx) = cT(x) for all x ∈ R

n and c ∈ R.

Fact: If T : R

n → R

m is a linear transformation, then T(0) = 0.

We’ve already met examples of linear transformations. Namely: if A is

any m × n matrix, then the function T : R

n → R

m which is matrix-vector

multiplication

T(x) = Ax

is a linear transformation.

(Wait: I thought matrices were functions? Technically, no. Matrices are lit-

erally just arrays of numbers. However, matrices define functions by matrix-

vector multiplication, and such functions are always linear transformations.)

Question: Are these all the linear transformations there are? That is, does

every linear transformation come from matrix-vector multiplication? Yes:

Prop 13.2: Let T : R

n → R

m be a linear transformation. Then the function

T is just matrix-vector multiplication: T(x) = Ax for some matrix A.

In fact, the m × n matrix A is

A =



T(e1) · · · T(en)



.

Terminology: For linear transformations T : R

n → R

m, we use the word

“kernel” to mean “nullspace.” We also say “image of T” to mean “range of

T.” So, for a linear transformation T : R

n → R

m:

ker(T) = {x ∈ R

n

| T(x) = 0} = T

−1

({0})

im(T) = {T(x) | x ∈ R

n

} = T(R

n