MAT223H5 Study Guide - Midterm Guide: Elementary Matrix, Linear Map, Catalan Shawm

• School: UTM
• Department: Mathematics
• Course: MAT223H5
• Professor: Marina Tvalavadze

Document Summary

You have 50 minutes to complete the test. No calculators, books or other aids are permitted. Find all values of the parameter a for which this matrix is noninvertible. Solution: a row echelon form of a is as follows: Notice that a row echelon form of a matrix is not unique. It follows from the invertible matrix theorem that a is noninvertible if and only if the number of pivots is less than 4. Since 1 + a2 is always nonzero, we have that a is noninvertible if and only if a = 0. Problem 2. (10 points) a linear transformation t : r3 (cid:55) r4 is given by. T (e1) = e1 + 5e3 + e4, T (e2) = e2 e3 + e4, A = [t (e1) t (e2) t (e3)] = 1 2 (b) in order for t to be one-to-one columns of a must be linearly independent. Equivalently, ax = 0 must only have the trivial solution.