Ch. 7 Linear Mappings

Question 1

Definition of a linear map

Answer 1

The linear map T:V –> W is linear if:

i) for all u and v in V, T(u + v) = T(u) + T(v
(ii) for all v in V and all λ in R T(λv) = λT(v)

Question 2

How to compose 2 linear maps S and T

Answer 2

Find the 2 matrices of the maps T(x) = Ax and S(y) =By

(S o T)x = (BA)x

Question 3

Definition of isomorphic

Answer 3

If T: V –> W be a bijective linear map of real vector spaces. Then V and W are isomorphic

Question 4

Image of the linear map T : V –> W

Answer 4

im(T) = {w ϵ W | there exists v ϵ V s.t w = T(v)}

Question 5

Kernal of the linear map T : V –> W

Answer 5

ker(T) = {v ϵ V | T(v) = 0}

Question 6

Rank of the linear map T : V –> W

Answer 6

rk(T) = dim(im(T))

Question 7

Nullity of the linear map T : V –> W

Answer 7

n(T) = dim(ker(T))

Question 8

Suppose the linear map T is given by matrix A with respect to the standard basis
How to find the matrix A with respect to a basis

Answer 8

Put the basis in a matrix P, find the inverse

The required matrix B = P^-1AP

Question 9

Suppose the linear map T is given by matrix A with respect to the standard basis
How to find the matrix A with respect to 2 bases

Answer 9

Put both the bases in matrices P and Q (decide P and Q based on dimensions)
Find the inverse of Q
The required matrix B = Q^-1AP

Question 10

Given a linear transformation eg T : R^3 –> R^3

How to find the matrix which represents T with respect to both copies of R^3

Answer 10

Find T(e1), T(e2), T(e3) and put them in a matrix