linear algebra

HOMEWORK 5

DUE MONDAY MAY 7TH AT 2PM

Your homework should be written on standard-sized paper, and loose sheets

must be stapled together.

(1) Consider the four vectors

a = (2, 1, 0), b = (1, 0,−1), c = (−1, 1, 1), d = (0,−1, 1),

in R3. Let U = spanR(a,b) and W = spanR(c,d). Find a basis for each of the subspaces U, W , and U ∩W , and hence use the dimensions of these three spaces to conclude that U + W = R3. Justify all of your answers.

(2) Find a basis for each of the following vector spaces, and hence compute their

dimension. (You may assume that each space uses the standard addition and

scalar multiplication for that type of vector space, except where specified.)

(a) V1 = {(x,y,z) ∈ R3 | 3x + 2y + z = 0}.

(b) V2 = {p(x) ∈ P5(R) | p′(x) has degree ≤ 2}. (p′(x) is the usual definition of derivative of polynomials.)

(c) V3 = C2, viewed as a vector space over R.

(3) Let T : R2 −→ P2(R) be the map defined by

T(a,b) = (a + b) + a2λx2,

where λ ∈ R. For what values of λ is T a linear map? Justify your answer.

(4) Let b ∈ F, and define a map T : F∞ −→ F∞ by

T(x1,x2,x3, . . . ) = (b,x1,x2, . . . ).

Prove that T is a linear map if and only if b = 0.

(5) Let T ∈ L(V,W) and suppose that v1, . . . ,vn ∈ V are such that Tv1,Tv2, . . . ,Tvn are linearly independent in W . Prove that v1, . . . ,vn are

linearly independent in V .

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