Oldprelim - Prelim 1 Math 294 4 questions total 100 points You may use anything that has been given in class or in the book as long as you show

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Prelim 1 Math 294 September 28, 2006 4 questions; total 100 points . You may use anything that has been given in class or in the book, as long as you show clearly what you are using . Calculators are neither needed nor permitted. Some pieces of formulas and deFnitions can be found on the bottom of the second side. 1. We are given a system of 4 linear equations in 6 unknowns in the form A±x = ± b , and told that the augmented matrix A ± b row reduces to 1 3 0 -1 1 0 1 0 0 1 1 2 0 2 0 0 0 0 0 1 3 0 0 0 0 0 0 0 (a) (10 points) ±ind all solutions to A±x = ± b . (b) (4 points) Let T : R p R q be the linear transformation given by T ( ±x )= A±x . Determine p and q . (c) (4 points) Determine rank( A ) and the dimension of image( T ). (d) (4 points) Is it true that ± b is an element of image( T )? Why or why not? (e) (4 points) Determine the dimension and a basis for the kernel of T . (f) (4 points) If A = ± ±v 1 | 2 | · · · | 6 ² , give a basis for image( T ). Do you have enough information to compute these vectors explicitly?
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This note was uploaded on 10/01/2008 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell University (Engineering School).

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Oldprelim - Prelim 1 Math 294 4 questions total 100 points You may use anything that has been given in class or in the book as long as you show

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