prelim1 fa06 - Prelim 1 Math 294 September 28, 2006 4...

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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 ² , give a basis for image( T ). Do you have enough information to compute these vectors explicitly? How would you do it, if so? (Do not do the calculations.) 2. Consider the matrix A = 1 2 0 0 1 2 1 2 1 . (a) (15 points) Find A - 1 and check your answer using matrix multi- plication. (b) (5 points) If T is the linear transformation defined by T ( ~x ) = A~x then determine the vector ~v so that T ( ~v ) = 1 1 1 .
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3. Let T be the linear transformation that reflects R 3 through the plane x - y + 2 z = 0. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! B B B B B B T ( ~x ) ~x (a) (10 points) Determine a basis B = { ~v 1 ,~v 2 ,~v 3 } for R 3 so that ~v 1 is orthogonal to the plane and ~v 2 ,~v 3 are on the plane. (b)
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prelim1 fa06 - Prelim 1 Math 294 September 28, 2006 4...

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