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Practice Prelim 1

# Practice Prelim 1 - Mathematics 2940 Practice Questions for...

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Mathematics 2940 Practice Questions for Prelim 1 Fall 2010 These are typical questions, perhaps more than you would expect on a single prelim. 1. Let A = 2 1 3 0 1 - 1 2 1 4 - 1 7 2 and b = 5 0 4 . (a) Find all solutions to the equation A x = b . (b) Find all solutions to the equation A x = 0 . (c) Find a basis for the nullspace and the column space of A, and state the dimension of each of these spaces. 2. Let A = 1 0 1 0 1 1 0 0 - 2 . (a) Show that A is invertible and find A - 1 . (b) Express A - 1 as a product of elimination matrices. (c) Use your answer to (a) to find a 3 × 3 matrix X such that XA = 1 0 1 0 1 0 1 0 1 . (d) Determine the LU or PLU factorization, as appropriate, of the matrix A + A T . 3. (a) Let W be the set of vectors in R 4 orthogonal (perpendicular) to all of the vectors (0 , 1 , - 2 , 3), (1 , 1 , - 4 , 2), and (2 , - 1 , 0 , - 4). Show that W is a subspace of R 4 . (b) If u , v , and w are any vectors in R 4 , show that there is a non-zero vector orthog- onal to all of them.

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