# Oct2011 mat224 test1 - ww w ox di a University of Toronto...

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University of Toronto Department of Mathematics MAT224H1F Linear Algebra II Midterm Examination October 25, 2011 S. Uppal Duration: 1 hour 50 minutes Last Name: Given Name: Student Number: Tutorial Group: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /10 2 /10 3 /10 4 /10 5 /10 6 /10 TOTAL /60 1 of 11 DownloaderID 24460 ItemID 15891 Downloader ID: 24460 Downloader ID: 24460 Downloader ID: 24460 Item ID: 15891 Item ID: 15891 Item ID: 15891 Item ID: 15891
[10] 1. Let T : M 2 × 2 ( R ) M 2 × 2 ( R ) be the linear transformation defined by T ( A ) = A + A T 2 . Find the matrix of T relative to the basis α = { 1 1 0 0 , 0 0 1 1 , 1 0 0 1 , 0 1 1 1 } for M 2 × 2 ( R ). SOLUTION: Let v 1 = 1 1 0 0 , v 2 = 0 0 1 1 , v 3 = 1 0 0 1 , and v 4 = 0 1 1 1 . Then T ( v 1 ) = 1 2 1 1 0 0 + 1 0 1 0 = 1 1 / 2 1 / 2 0 = 1 2 (3 v 1 + 3 v 2 v 3 2 v 4 ) , T ( v 2 ) = 1 2 0 0 1 1 + 0 1 0 1 = 1 2 0 1 1 2 = 1 2 ( v 1 v 2 + v 3 + 2 v 4 ) , T ( v 3 ) = 1 2 1 0 0 1 + 1 0 0 1 = 1 0 0 1 = v 3 , T ( v 4 ) = 1 2 0 1 1 1 + 0 1 1 1 = 0 1 1 1 = v 4 . Therefore, [ T ] αα = 3 / 2 1 / 2 0 0 3 / 2 1 / 2 0 0 1 / 2 1 / 2 1 0 1 1 0 1 . 2 of 11 DownloaderID 24460 ItemID 15891
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[10] 2. Let T : P 2 ( R ) P 2 ( R ) be the linear transformation defined by T ( a + bx + cx 2 ) = ( 2 b + 11 c ) + ( 2 a + c ) x + (3 a b + 4 c ) x 2 . Find bases for the kernel and image of T .