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oldprelim - Prelim 1 Math 294 4 questions total 100 points...

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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 definitions 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) Find 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 | v 2 | · · · | v 6 , give a basis for image( T ). Do you have enough information to compute these vectors explicitly?
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