Consider the subspace W = s pan − 1 a Use the...

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University of Toronto Department of Mathematics MAT223H1F Linear Algebra I Midterm Examination II November 25, 2010 H. Kim, F. Murnaghan, B. Rowe, S. Uppal Duration: 1 hour 20 minutes Last Name: Given Name: Student Number: Tutorial Code: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /10 2 /10 3 /10 4 /10 5 /5 6 /5 TOTAL /50 1 of 9 DownloaderID 21140 ItemID 3250
1. Consider the subspace W = s pan 1 (a) Use the Gram-Schmidt process to find an orthonormal basis for W . braceleftBigg 1 0 0 , 1 1 0 0 , 0 1 0 1 bracerightBigg of R 4 . (b) Find the orthogonal projection of the vector Downloader ID: 21140 Downloader ID: 21140
|| u 2 || = 6 Downloader ID: 21140 Item ID: 3250
1 2 of 9 Downloader ID: 21140 Item ID: 3250 Dow Downloader ID: 21140 Item ID: 3250
Let u 3 = v 3 ( u 2 ,v 3 ) ( u 2 ,u 2 ) u 2 ( u 1 ,v 3 ) ( u 1 ,u 1 ) u 1 = 1 3 2 2 0 2 , and let e 3 = u 3 || u 3 || = 3 3 1 1 0 1 . Then { e 1 ,e 2 ,e 3 } form an orthonormal basis of W . (b) Let v = 1 0 0 1 . Then proj W ( v ) = ( e 1 ,v ) e 1 + ( e 2 ,v ) e 2 + ( e 3 ,v ) e 3 = 0 e 1 + 6 3 e 2 + 2 3 3 e 3 = 1 0 0 1 Remark: Part (b) shows that the vector v is actually in W . 3 of 9 DownloaderID 21140 ItemID 3250

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