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University of Toronto Department of Mathematics MAT223H1F Linear Algebra I Midterm Examination October 22, 2009 H. Kim, S. Kudla, F. Murnaghan, S. Uppal Duration: 1 hour 50 minutes Last Name: Given Name: Student Number: Tutorial Code: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /6 2 /10 3 /10 4 /10 5 /10 6 /10 7 /4 8 /5 TOTAL /65 1 of 9 utoronto.studentbuddy.com DownloaderID 21140 ItemID 1792 Downloader ID: 21140 Downloader ID: 21140 Downloader ID: 21140 Item ID: 1792 Item ID: 1792 Downloader ID: 21140
[6] 1. Find all solutions of the homogeneous linear system Ax = 0, where A is a 3 × 6 matrix whose reduced row echelon form is 1 2 0 3 0 5 0 0 1 4 0 6 0 0 0 0 1 1 . Express your answer in parametric form. Solution The reduced row echelon form of A has 3 leading ones, so we will assign three param- eters to non-leading variables : x 2 = s, x 4 = t, x 6 = u. From the last row we can see that x 5 = u. The second row sais that x 3 + 4 x 4 + 6 x 6 = 0 x 3 = 4 t 6 u. The third row means that x 1 + 2 x 2 + 3 x 4 + 5 x 6 = 0 x 1 = 2 s 3 t 5 u. Thus all the solutions x = ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ) T of the homogeneous linear system Ax are of the form ( 2 s 3 t 5 u,s, 4 t 6 u,t, u,u ) T = s ( 2 , 1 , 0 , 0 , 0 , 0) T + t ( 3 , 0 , 4 , 1 , 0 , 0) T + u ( 5 , 0 , 6 , 0 , 1 , 1) T where s,t,u are real numbers. 2 of 9 utoronto.studentbuddy.com DownloaderID 21140 ItemID 1792 Item ID: 1792 Downloader ID: 21140 Item ID: 1792 Item ID: 1792 Downloader ID: 21140 Downloader ID: 21140 Downloader ID: 21140 Downloader ID: 21140 Item ID: 1792 Item ID: 1792 Downloader ID: 21140 Item ID: 1792 Downloader ID: 21140 Item ID: 1792 Item ID: 1792
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