MAT224 ASSIGNMENT 2
DUE BY FRIDAY JUNE 5, 2020, 11:59 PM
Each question is worth 5 marks.
Question 1. Define T : Pn → R via T(p(x)) = ak, where p(x) = a0+a1x+…+anxn. Prove that dim(kerT) = n.
Question2.LetV→T W→S VbefunctionssuchthatTS=1W andST=1V.IfTislinear,provethatSislinear. n
Question 3. Given {v1,…,vn} in a vector space V, define T : Rn → V by T(a1,…,an) = akvk. Show that T k=1
is linear, and that ker T = {0} if and only if {v1, . . . , vn} is an independent set.
Question 4. Suppose T : P2 → P3 is a linear transformation with T(x2) = x3, T(x + 1) = 2, and T(x − 1) = 2x. Find the following.
(a) T (x2 + 2x + 2).
(b) T(x).
Question 5. Suppose N is an n × n matrix and Nk = 0 (the zero matrix) for some natural number k. Prove that T :Mnm →Mnm givenbyT(A)=A−NAisanisomorphism.
n k=0
1