MAT224 ASSIGNMENT 4
DUE BY FRIDAY AUGUST 7, 2020, 11:59 PM
Remark: 5 of the following questions will be marked, but you must do all of them. Each question that is marked is out of 5 for a total of 25 marks.
Question 1. Let D : P3 → P2 be the differentiation map given by D(p(x)) = p′(x). Find the matrix of D corresponding to the basis B = {1,x,x2,x3} and E = {1,x,x2} (that is, find MEB(D)), and use it to compute D(a+bx+cx2 +dx3).
Question 2. In P3 find PD←B if B = {1,x,x2,x3} and D = {1,(1 − x),(1 − x)2,(1 − x)3}. Then express p(x) = a+bx+cx2 +dx3 as a polynomial in powers of (1−x).
Question 3. Define T : R2 → R2 via T (a, b) = (a − b, 2b − a). Find det T , tr T , and CT (x).
Question 4. Let T : V → V be a linear operator satisfying T2 = T. Define U1 = {v ∈ V : T(v) = v} and
U2 = {v ∈ V : T(v) = 0}. Prove that V = U1 ⊕ U2.
Question 5. Suppose U and W are subspaces of V , dim V = n, dim U + dim W = n, and U ∩ W = {0}. Prove that V=U⊕W.
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