MATH 223, 2017W

Homework set 4 — due October 11th

Problem 1. (i) Let V1, V2, V3 be three real vector spaces and let ϕ : V1 → V2 and ψ : V2 → V3 betwo linear transformations. Prove that their composition

ψ ◦ ϕ : V1 −→ V3

x 7−→ (ψ ◦ ϕ)(x) = ψ(ϕ(x))

is a linear transformation.(ii) Assume that V = W1 ⊕W2 and let

P : V −→ V

x = x1 + x2 7−→ P (x) = x1

Prove that P ◦ P = P . Characterize the kernel N (P ) and the range R(P ).(iii) Let Q : V → V be a linear transformation such that Q◦Q = Q. Prove that V = N (Q)⊕R(Q)(Hint: x = x−Q(x) +Q(x))

Problem 2.Let V be a finite-dimensional vector space.(i) Let {e1, . . . en} and {f1, . . . fn} be two bases of V . Prove that there is a unique linear transfor-mation ϕ : V → V such that ϕ(ei) = fi.(ii) Let ψ : V → V be a linear transformation such that N (ψ) = {0}. Prove that {ψ(e1), . . . ψ(en)}is another basis of V .

Problem 3. For any matrix M ∈Mp×n, we define its transpose MT ∈Mn×p by (MT )i,j = Mj,i.Let A,B,C,D be the matrices

A =

(1 23 1

), B =

(1 0 1−2 −1 3

), C =

(0 2 1−1 −2 −3

), D =

−121

.

(i) Identify all of the following products that are defined and give the size of the resulting matrix:CTA, CD ,AB ,BA ,BTC ,DTB ,CTAB.(ii) Compute

A(B + C), A(BD), CTB, (AB)D, DTD.

Problem 4.For any θ ∈ [0, 2π), let

R(θ) =

(cos θ − sin θsin θ cos θ

)∈M2×2.

(i) Check that R(θ) is the matrix of a rotation in R2 around the origin by an angle θ, with respectto the canonical basis {( 1

0 ) , ( 01 )}.

(ii) Use the fact that the composition of a rotation by θ followed by a rotation by ω is a rotationby θ + ω to derive the trigonometric identities for sin(θ + ω) and cos(θ + ω).Hint: The composition of maps is given by the product of the corresponding matrices(iii) Find the matrix inverse of R(θ).