Quiz - 2 Problems on Linear Algebra- Define Linear Transformation | MATH 423, Quizzes of Mathematics

Download Quiz - 2 Problems on Linear Algebra- Define Linear Transformation | MATH 423 and more Quizzes Mathematics in PDF only on Docsity! Math 423 The real quiz 5 Name: J. Zinn ID# 1. Define the linear transformation (or matrix) M : R6 −→ R6 by M =  1 0 −1 1 −2 0 1 0 −1 0 −1 0 0 1 −1 0 0 0 0 0 0 1 −1 0 0 0 0 1 −1 0 0 0 0 0 0 0 . Assume (I’m giving you this information) that R(M2) = sp(e1 + e2 + e3) = sp(M2e1),Me1 = e1 + e2, etc. ker(M) = sp(e1 + e4 + e5, e2 − e3 − e4 − e5, e6). (a) Show that M is nilpotent. (b) Find the basis which gives the Jordan form for the matrix, M . (c) Find the Jordan matrix, J , which is similar to M . 2. Prove that if M ⊆ M +N and β = {b1, b2, . . . , bk} is a basis for M , then there exists a basis for M +N , which extends β, say, γ = {b1, b2, . . . , bk, c1, . . . , cl} such that the new elements, {cj}lj=1, are in N . 1