Linear algebra hw due 3 hrs

HOMEWORK 7

DUE WEDNESDAY MAY 30TH AT 2PM

Your homework should be written on standard-sized paper, and loose sheets must be stapled together.

(1) Let v1 = (1,−1) and v2 = (−1, 2), and note that {v1,v2} is a basis for R2. (You do not have to prove this.) Let S,T : R2 −→ R3 be defined on this basis by

S(v1) = (1, 0,−1), S(v2) = (−2, 0, 2) T(v1) = (−1, 0, 1), T(v2) = (5, 1, 3)

and extended by linearity. What is: (a) im(S + T)?

(b) ker(S + T)?

(c) (3S + T)(2, 3)?

(2) For each of the following pairs of spaces, decide whether they are isomorphic or not. If yes, give an explicit isomorphism between the two spaces. If not, find an injective linear map from one to the other and verify that it is not surjective. In each case, justify your answer by proving that your map is linear and has the properties you claim it does.

(a) C2 and P3(C) (as vector spaces over C)

(b) C2 and P3(R) (as vector spaces over R)

(c) spanR(x,x 3,x4) (viewed as a subpsace of P4(R) over R) and

{(x,y,z) | x + y = z} (viewed as a subspace of R3 over R)

(d) spanQ((1, 2), ( 1 2 , 1)) (viewed as a subspace of Q2 over Q) and Q.

(3) Let U = R be the vector space over R with the usual addition and scalar multiplication. Let V = R, with an addition � and scalar multiplication · defined by

a � b := a + b− 2; λ ·a := λa− 2λ + 2.;

for all a,b ∈ V and all λ ∈ R. This is also a vector space over R (see HW2). Prove that V ∼= U by finding an explicit isomorphism between the two vector spaces. (Your answer should include a proof that the map you wrote down is indeed an isomorphism.)

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2 DUE WEDNESDAY MAY 30TH AT 2PM

(4) Let V be a finite dimensional vector space over a field F. Find a basis for the vector space L(F,V ) and hence show that V ∼= L(F,V ).

[Hint: let v ∈ V . If T,S ∈ L(F,V ) are such that T(1) = λv and S(1) = µv, how are T and S related?]

(5) Let V be a finite dimensional vector space over a field F. Find a basis for the vector space L(V,F) and hence show that V ∼= L(V,F).

[Hint: what are the simplest non-zero elements of L(V,F) that you can think of?]