November 14, 2020

Math 108A (section 300)

Review Problems for Midterm 2

(Will not be collected.)

The problems are taken from the Axler text, but they are also repro-

duced here.

1.) Problem 3.B.9: Suppose T ∈ L(V,W ) is injective and v1, . . . , vn is

linearly independent in V . Prove that Tv1, . . . , T vn is linearly indepen-

dent in W .

2). Problem 3.B.14: Suppose U is a 3-dimensional subspace of R8 and

that T is a linear map from R8 to R5 such that nullT = U . Prove that

T is surjective.

3). Problem 3.C.2: Suppose that D ∈ L(P3(R),P2(R)) is the differen-

tiation map defined by Dp = p′. Find a basis of P3(R) and a basis of

P2(R) such that the matrix of D with respect to these bases is1 0 0 00 1 0 0

0 0 1 0

.

4). Problem 3.D.3: Suppose V is finite-dimensional, U is a subspace

of V , and S ∈ L(U, V ). Prove that there exists an invertible operator

T ∈ L(V ) such that Tu = Su for every u ∈ U if and only if S is

injective.

5). Problem 3.D.8: Suppose V is finite-dimensional and T : V → W is a

surjective linear map of V onto W . Prove that there is a subspace U of

V such that T |U is an isomorphism of U onto W . (Here T |U means the

function T restricted to U . In other words, T |U is the function whose

domain is U , with T |U defined by T |U(u) = Tu for every u ∈ U .)

6). Problem 3.D.11: Syppose V is finite-dimensional and S, T, U ∈

L(V ) and STU = I. Show that T is invertible and T−1 = US.

7). Problem 3.E.6: For n a positive integer, define V n by

V n = V × · · · × V︸ ︷︷ ︸

n times

.

Prove that V n and L(Fn, V ) are isomorphic vector spaces.

8). Problem 3.E.11: Suppose v1, . . . , vm ∈ V . Let

A = {λ1v1 + · · ·+ λmvm : λ1, . . . , λm ∈ F and λ1 + · · ·+ λm = 1}.

(a) Prove that A is an affine subset of V .

(b) Prove that every affine subset of V which contains v1, . . . , vm

also contains A.

(c) Prove that A = v + U for some v ∈ V and some subspace U of

V with dimU ≤ m− 1.

9). Problem 3.E.17: Suppose U is a subspace of V such that V/U is

finite-dimensional. Prove that there exists a subspace W of V such

that dimW = dimV/U and V = W ⊕ U .

10). Problem 3.F.4: Suppose V is finite-dimensional and U is a sub-

space of V such that U 6= V . Prove that there exists ϕ ∈ V ′ such that

ϕ(u) = 0 for all u ∈ U but ϕ 6= 0.

11). Problem 3.F.9: Suppose v1, . . . , vn is a basis of V and ϕ1, . . . , ϕn

is the corresponding dual basis of V ′. Suppose ψ ∈ V ′. Prove that

ψ = ψ(v1)ϕ1 + · · ·+ ψ(vn)ϕn.

12). Problem 3.F.11: Suppose A is an m×n matrix with A 6= 0. Prove

that the rank of A is 1 if and only if there exist (c1, . . . , cm) ∈ Fm and

(d1, . . . , dn) ∈ Fn such that Aj,k = cjdk for every j = 1, . . . ,m and

every k = 1, . . . , n.

13). Problem 3.F.18: Suppose V is finite-dimensional and U ⊂ V .

Show that U = {0} if and only if U0 = V ′.

14). Problem 4.4: Suppose m and n are positive integers with m ≤ n,

and suppose λ1, . . . , λm ∈ F. Prove that there exists a polynomial

p ∈ P(F) with deg p = n such that 0 = p(λ1) = · · · = p(λm) and such

that p has no other zeros.

15). Problem 4.8: Define T : P(R)→ RR by

Tp =

{

p−p(3)

x−3 if x 6= 3

p′(3) if x = 3.

Show that Tp ∈ P(R) for every polynomial p ∈ P(R) and that T is a

linear map.

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