chuang_exam1

chuang_exam1 - B is indeed P 2 . Do not tacitly choose a...

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MATH 107: EXAM 1 No calculators allowed. Your work must be clearly written in order to receive credit. Time: 50 minutes (1) (21 points) Let M = 2 1 - 2 - 1 3 - 3 4 - 2 1 b = 1 1 1 (a) (5 points) Compute the determinant of M by cofactor expansion along the second row. (b) (5 points) Compute the determinant of M by row-reducing M to an upper-triangular matrix. (c) (6 points) Solve M x = b by Cramer’s Rule. (d) (5 points) Find a LU-decomposition for M . Hint: Use your work from part (b). (2) (15 points) The set B = { x 2 - x + 2 , 3 x 2 + x - 1 , 4 x 2 + 2 x - 3 } is a basis for the vector space R [ x ] 2 = P 2 of real polynomials of degree at most 2. (a) (10 points) Verify that the span of
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Unformatted text preview: B is indeed P 2 . Do not tacitly choose a basis for the vector space. (b) (5 points) Find [ x 2 ] B . (3) (28 points) Let A = -1 1 1 3-1-3-2 1-3-7-7-2 3 5 8 (a) (10 points) Find a row- echelon form for A . (b) (15 points) Find a basis for each of the subspaces NS ( A ), CS ( A ), and RS ( A ). Clearly label which basis is for which subspace. (c) (3 points) Explain geometrically why the basis for NS ( A ) and the transpose of that for RS ( A ) together constitute a basis for R 4 . 1...
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