1.8 Intro to Linear Transformation/ 1.9 Matrix for a Linear Transformation

Question 1

Transformation

Answer 1

function T: Rn –> Rm

Question 2

domain of transformation for T: Rn –> Rm

Answer 2

Rn

Question 3

codomain of transformation for T: Rn –> Rm

Answer 3

Rm

Question 4

matrix transformation

Answer 4

a transformation T: Rn –> Rm where T(x) = Ax and A is an m x n matrix

Question 5

A transformation T: Rn –> Rm is a linear transformation if:

Answer 5

( 1 ) for all vectors x, y belong to Rn, T(x+y)=T(x)+T(y)

2 ) for all vector x belong to Rn and for all c belong to R, T(cx)=cT(x

Question 6

Image

Answer 6

Let T: Rn –> Rm be a transformation. The image of x belong to Rn is T(x)

IMAGE = OUTPUT

Question 7

Image of T

Answer 7

the set ImT = {T(x) such that x belong to Rn}

range does not equal codomain

Question 8

Linear transformation and matrix transformation relation?

Answer 8

a linear transformation T: Rn –> Rm is EQUIVALENT to the matrix transformation x|–> Ax where A is an m x n matrix

Question 9

standard matrix

Answer 9

[ T(e1) T(e1) …. T(en) ]

Question 10

for a transformation T: Rn –> Rm:

Answer 10

( 1 ) if the image is Rm then we say T is onto or subjective

( 2 ) if T(x1) = T(x2) implies x1=x2 then we say T is one to one or injective

Question 11

If T is onto then

Answer 11

• T(x)=b has a solution for all b belonging to Rm

- A HAS A PIVOT IN EVERY ROW

Question 12

If T is one to one then

Answer 12

• Ax = b has at most one solution

- A HAS A PIVOT IN EVERY COLUMN

Question 13

Suppose T: Rn –> Rm is a linear transformation with standard matrix A

Answer 13

( 1 ) T is onto if and only if the columns of A span Rm

( 2 ) T is one to one if and onl if the columns of A are linearly independent