MATH235 Study Guide - Final Guide: Empty Set

• School: University of Waterloo
• Department: Mathematics
• Course: MATH235
• Professor: Ghazal Geshnizjani

Document Summary

Assignment 1 solutions: find a basis for the four fundamental subspaces of a = . Since rank(at ) = rank(a) = 3, we know by the dimension theorem that dim null(at ) = So, a basis for null(at ) is the empty set: let a = (cid:20)0 1. 1 0(cid:21) and de ne l : m2 2(r) m2 2(r) by l (x) = ax xa. (a) prove that l is linear. Solution: let x, y m2 2(r), and s, t r. then. L[sx + ty ] = a(sx + ty ) (sx + ty )a. = sax + tay sxa ty a. = s(ax xa) + t(ay y a) 1 3(cid:21). (c) find a matrix c m2 2(r) such that l(c) = (cid:20)3 1. Solution: we need to nd c = (cid:20)a b a d b c(cid:21) 1 3(cid:21) = l (c) = (cid:20) c b d a (cid:20)3 1 c d(cid:21) such that.