4

Question 1

Suppose that V and W are (real) vector
spaces. A function T → V ! W is linear if for all u, v ∈ V and all α ∈ R, …

Answer 1

1. T(u + v) = T(u) + T(v) and
2. T(αu) = αT(u).
   T is said to be a linear transformation (or linear mapping or linear function).

Equivalently, T is linear if for all u, v ∈ V and α, β ∈ R,
T(αu +βv) = αT(u) + βT(v):

Question 2

Suppose that T is a linear transformation from vector space V to vector space W. Then the range, R(T), of T is

Answer 2

R(T) = {T(v) : v ∈ V}

Question 3

Suppose that T is a linear transformation from vector space V to vector space W. Then the null space, N(T), of T is

Answer 3

N(T) = {v ∈ V : T(v) = 0};
where 0 denotes the zero vector of W.
The null space is often called the kernel, and may be denoted ker(T) in some texts.
Of course, for any matrix, R(T_A) = R(A), and N(A_T) = N(A).

Question 4

If V and W are both finite dimensional, then so are R(T) and N(T). We define the rank of T, rank(T) to be … and the nullity of T, nullity(T), to be … .

Answer 4

rank(T) is dim (R(T))

nullity of T, nullity(T) = dim (N(T)).

Question 5

Suppose that T is a linear transformation from the finite-dimensional vector space V to the vector space W (not necessarily finite-dimensional.
Then …

Answer 5

rank(T) + nullity(T) = dim(V)

Question 6

For an m x n matrix A, if T = T_A, then T is a linear transformation from V = Rⁿ to W = R^m, and rank(T) = rank(A), nullity(T) = nullity(A), so…

Answer 6

rank(A) + nullity(A) = n

Question 7

if we let P_B be the matrix whose columns are the basis vectors (in order), P_B = (x₁x₂ …x_n), then for any x ∈ Rⁿ, x =

Answer 7

x = P_B [x]_B

The matrix P_B is invertible (because its columns are linearly independent, and hence its rank is n). So we can also write [x]_B = P_B^-1 x

Question 8

Suppose that T : Rⁿ → R^m is a linear transformation, that B is a basis of Rⁿ and B’ is a basis of R^m.

Let P_B and P_B’ be the matrices whose columns are, respectively, the vectors of B and B’.

Then the matrix representing T with respect to B and B’ is given by A_T [B,B’] = P_B’^-1 A_TP_B;
where AT = (T(e₁) T(e₂) … T(e_n))
So, for all x, [T(x)]_B’ = …

Answer 8

[T(x)]_B’ = PB^‘-1 A_TP_B[x]_B

Question 9

Suppose that T : Rⁿ → Rⁿ is a linear transformation and that B = {x₁; x₂… x_n} is some basis of Rⁿ. Let
P = (x₁ x₂… x_n) be the matrix whose columns are the vectors of B.

Then for all x ∈ Rⁿ,
[T(x)]_B = P^-1A_TP[x]_B;
where A_T is the matrix corresponding to T,
A_T = (T(e₁) T(e₂) … T(e_n)).
In other words, A_T [B,B] = …

Answer 9

A_T [B,B] = P^-1A_TP

Question 10

The relationship between the matrices A_T [B,B] and A_T is an important one in the theory of linear algebra.

If there is an invertible (nonsingular) matrix P such that

B = P^-1AP, we say that two square matrices A and B are

Answer 10

similar