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# test3b - Version 2 1 Solutions to Test 3 MATH 1104F Winter...

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Version 2 1 Solutions to Test 3 - MATH 1104F - Winter 2010 Version 2 PART I: Multiple choice questions (3 points each) Choose and circle only one answer. No partial marks here. No justification is required. 1. Let T : R 2 R 2 be the linear transformation given by T ( x, y ) = ( - y, x ). Then the inverse of T is: ( a ) T - 1 ( x, y ) = ( - x, y ) ( b ) T - 1 ( x, y ) = ( x, - y ) ( c ) T - 1 ( x, y ) = ( - y, x ) ( d ) T - 1 ( x, y ) = ( y, - x ) Solution: (d) 2. Let S 1 = { (1 , 2 , 0) , ( - 1 , 1 , 0) } , S 2 = { ( a + b, a - b, 1): a and b are real numbers } , S 3 = Span { (1 , 0 , - 1) , (2 , - 1 , 1) } , and S 4 = Col 1 0 1 0 1 1 1 1 0 . The following are subspaces of R 3 : ( a ) S 1 and S 2 only ( b ) S 1 and S 4 only ( c ) S 2 and S 3 only ( d ) S 3 and S 4 only Solution: (d) 3. Let A be an n × n matrix. Consider the following statements: (i) A is an invertible matrix. (ii) The columns of A span R n . (iii) rank A > 0. (iv) The linear transformation x 7→ Ax is one-to-one. Which statement is NOT equivalent to the others? (a) (i) (b) (ii) (c) (iii) (d) (iv) Solution: (c) 4. Let A , B and C be 4 × 4 matrices such that: B is obtained from A by interchanging Row 1 and Row 2; C is obtained from

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test3b - Version 2 1 Solutions to Test 3 MATH 1104F Winter...

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