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MATH235: Linear Algebra 2

Crowdmark Assignment 2

Fall 2022

Problems

Q1. Let B = { v(#»)1 , . . . , v(#»)n} be an ordered basis for the vector space V , and let [ ]B : V → Fn  be the coordinate map that sends a vector x(#») ∈ V to the coordinate vector [ x(#»)]B .

(a) Prove that [ ]B  is linear.

(b) Determine the rank and nullity of [ ]B .

[Note: Both of these results occur in the notes as unproved theorems. Your solution cannot just “quote”the theorems. You must supply the proofs.]

Q2. Let L : M2×2(R) → P2(R) be a linear map.

(a) Prove that dim(Ker(L)) cannot be zero.

(b) Suppose that L is defined by

L ([ c(a)   d(b)]) = (a − 3b + d) + (−2a + 6b + 2c + 2d)x + (2a − 6b + 2d)x2 .

(i) Find a basis for Range(L).

(ii) Find a basis for Ker(L).

Q3. For α ∈ F, consider the evaluation map  evα : Pn(F) → F defined by evα(p(x)) = p(α).  For example, ev3 (2 − x + 4x3 ) = 2 − 3 + 4(3)3 .

(a) Prove that evα  is a linear map.

(b) Find a basis for Range(evα) and determine rank(evα).

(c) Write down a polynomial p(x) in Ker(evα). No justification required.

(d) For any α ∈ F, prove that the set of polynomials Uα  = {p ∈ Pn(F) : p(α) = 0} is a subspace of Pn(F). [Hint:  Don’t work too hard.]

(e) Determine dim(Uα).  [Hint:  If you solved (d) the “easy”way, then you won’t have to

work too hard here either.]

Q4. Let L : V → W be a linear map and let B = { v(#»)1 , . . . , v(#»)n} be a basis for V .

(a) Prove that if {L(v(#»)1 ), . . . ,L(v(#»)n)} is linearly independent, then nullity(L) = 0.              (b) Prove that if nullity(L) = 0 and dim(W) = n, then C = {L(v(#»)1 ), . . . ,L(v(#»)n)} is a basis

for W .