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PROBLEM SET FOR MATH 341 (LINEAR ALGEBRA), FALL 2023

5. Coordinates

Due October 23, 2023 by 11:59pm

For each of the following problems, show all work and fully justify your answer/s.

One check+/check/check- can be earned for each of the numbered sub/questions (1(a), 1(b), 1(c), 2(a), 2(b), 2(c), 3, 4).

(1) Consider the following n-by-n matrices A. Given the indicated basis β of R n, use the change-of-basis formula to find [TA] ββ , where TA : R n → R n is the linear map given by TA(⃗v) = A⃗v.

(Note: if β0 denotes the standard basis of R n, then A = [TA] β

(2) For each of the following linear maps T : Rn → Rn and the indicated basis β of R n find [T] ββ .

Then, using the change-of-basis formula, find an expression for T(x, y) when n = 2, or for T(x, y, z) when n = 3.

(a) For m ∈ R, let T : R 2 → R 2 be the map that reflects a vector across the line y = mx and β = ((1, m),(−m, 1)).

(b) For m ∈ R, let T : R 2 → R 2 be the map that projects a vector to the nearest point on the line y = mx and β = ((1, m),(−m, 1)).

(c) For a, b ∈ R, let T : R 3 → R 3 be the map that projects a vector to the closest point on the plane defined by z = ax + by, and β = ((1, 0, a),(0, 1, b),(−a, −b, 1)).

(3) Prove that β = (1 + x + x 2 , 1 − x, 1 + x) is a basis for P2(x).

Given a polynomial f(x) = a + bx + cx2 , find [f]β.

If D: P2(x) → P2(x) is the linear map that takes a polynomial to its derivative, find [D] ββ and [D] β0β, where β0 = (1, x, x2 ).

(4) Let A be an invertible n-by- n matrix. Let β0 be the standard basis of R n.

Let β1 be the set of column vectors of A. Show that β1 is a basis of R n and that A = [id]β0β1 .