# Exam2A - (a Find a basis for Col A(b Find a basis for Nul...

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M340L EXAM 2A SPRING, 2008 Dr. Schurle Your name: Your UTEID: Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones, . . . . 1. (8 points) Use a cofactor expansion to compute the following determinant and simplify your result. 1 - x - 2 3 0 3 - x 1 2 5 2 - x

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YOUR SCORE: /70 2. (8 points) Determine if the set of all polynomials of the form p ( t ) = a + bt + ( a + b ) t 2 is a subspace of P n for an appropriate value of n . Justify your answer. 3. (8 points) (a) If the null space of an 8 × 7 matrix A is 3-dimensional, what is the dimension of the column space of A ? (b) If the dimension of the row space of a 15 × 18 matrix B is 11, what is the dimension of the null space of B ?
4. (12 points) The matrix A = 6 12 0 24 1 8 16 1 34 1 5 10 1 22 0 5 10 0 20 1 is row equivalent to the matrix B = 1 2 0 4 0 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 . (a) Find a basis for Col A . (b) Find a basis for Nul

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Unformatted text preview: . (a) Find a basis for Col A . (b) Find a basis for Nul A . (c) Find a basis for the row space of A . (d) What is the dimension of Col A ? (e) What is the dimension of Nul A ? (f) What is the rank of A ? 5. (6 points each) (a) Use coordinate vectors to show that B = { 1 , t-1 , ( t-1) 2 } is a basis for the vector space of polynomials of degree at most 2. (b) Continuing from (a), ﬁnd q ( t ) if [ q ] B = 1 3 2 . (c) Continuing from (a), ﬁnd [ r ] B if r ( t ) = 1 + 3 t + 2 t 2 . 6. (8 points) Let D = { d 1 , d 2 , d 3 } and G = { g 1 , g 2 , g 3 } be bases for a vector space V , and suppose g 1 = 3 d 1-2 d 2 + d 3 , g 2 = d 1 + d 2 + 2 d 3 , and g 3 =-d 1 + 2 d 2 + d 3 . (a) Find the change-of-coordinates matrix from G to D . (b) Find [ x ] D if x = 2 g 1-3 g 2 + g 3 . 7....
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