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223final20071

223final20071 - Faculty of Arts and Science University of...

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Faculty of Arts and Science University of Toronto Final Examination , April/May 2007 MAT223H1S Linear Algebra I C. Beltran, G. Naot, R. Roeder, S. Uppal Duration: 3 hours Last Name: Given Name: Student Number: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /8 2 /8 3 /12 4 /12 5 /8 6 /16 7 /8 8 /8 TOTAL /80 1 of 13

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For each of the statements below, decide if it is true or false. Indicate your answer by shading in the box corresponding to your choice. Justify your answer by providing an appropriate proof or counter example. [4] 1(a) Let U be a subspace of R n and suppose X and Y are two non-zero vectors in R n . Then proj U ( X ) and proj U ( Y ) are orthogonal. true false [4] 1(b) Let T : R n R m be a linear transformation. Then T is onto if and only if dim ( ker ( T )) = n - m . true false 2 of 13
For each of the statements below, decide if it is true or false. Indicate your answer by shading in the box corresponding to your choice. Justify your answer by providing an appropriate proof or counter example.

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