m526t2_s10_pract

# m526t2_s10_pract - pr.4(15 pts Let A be a 7 by 9 matrix...

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Math 526 Practice Test 2 March 15, 2010 pr 1. (25 points) Let A = 0 7 - 1 3 1 1 15 - 9 9 . (a) Find the L - U - P factorization of A . (b) Use the L - U - P factorization of A to solve 0 7 - 1 3 1 1 15 - 9 9 x 1 x 2 x 3 = 16 11 33 .

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pr.2 (10 pts.) Let A be an n x m matrix, and b - a nonzero vector. For each of the following answer True or False. (a) The set of all vectors x = x 1 x 2 x 3 with x 1 + 3 x 2 + 5 x 3 = 0 is a vector space. (b) The set of all vectors x = x 1 x 2 x 3 such that x 1 + x 2 + x 3 = 3 and x 1 - 2 x 2 + 3 x 3 = 5 is a vector space. (c) The set of all linear combinations of the columns of A is a vector space. (d) The null space of the matrix A is a vector space. (e) The set of all 3x3 matrices is a vector space. pr.3 (25 pts.) Let A = 2 8 - 4 6 3 12 - 6 6 . (a) Find an echelon form of A .
(b) Find the reduced echelon form of A . (c) Which are the pivot (basic) variables? Which are the free variables? (d) Find a basis for the column space. (e) Find a basis for the null space. (f) What is the dimension of the null space? What is the dimension of the row space?

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Unformatted text preview: pr.4 (15 pts.) Let A be a 7 by 9 matrix with rank 4. Find (a) The number of pivot variables (b) The number of free variables (c) How many solutions does the system of equations Ax = have? (d) dim( N ( A T )) = pr.5 (10 pts.) Let A be a 3x2 matrix with rank 2. (a) How many pivots does A have? (b) Let R be the reduced echelon form of A . Which columns of R are the pivot columns? (c) Which rows of R are the pivot rows? (d) What are the values of the pivots of R ? (e) What are the values of the elements of R which are not pivots? (f) Write down the matrix R . pr.6 (20 pts.) Find the complete solution (also called general solution) to 1-2 3 1 3-7 11 3 4-9 14 4 x 1 x 2 x 3 x 4 = 4 14 18...
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