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Unformatted text preview: MATH 20F: LINEAR ALGEBRA, WINTER 2007 MIDTERM 2 PRACTICE PROBLEMS Note: This is not a practice exam; you do not have to be able to do these problems all in 50 minutes. (1) Find the inverse of the matrix A below or show that A is not invertible. A = 1 2 3 2 5 3 1 0 8 . (2) Determine if the set { x 3 1 , x 2 x 2 , 2 x 3 + x 2 + 1 } of polynomials is linearly independent. (3) The matrices A = 1 3 1 3 1 2 6 1 5 6 3 9 2 8 7 1 3 2 5 and B = 1 3 0 2 5 1 1 4 are row equivalent. (a) What is the rank of A ? (b) Find bases for Col ( A ) and Nul ( A ). (4) The maximal number of linearly independent rows of a 5 × 6 matrix A is 3. What is the dimension of its null space? (5) Suppose A = " 5 4 2 3 # and B = " 7 3 2 1 # . Define maps L : R 2 → R 2 and T : R 2 → R 2 by L ( x ) = A x and T ( x ) = B x . Is there a linear map 1 S : R 2 → R 2 such that S ( T ( x )) = L ( x ) for every x in R 2 ? If so, find an explicit formula for S ( x ) and if not, explain why not. (6) Repeat problem 5 with A = " 4 2 8 4 # and B = " 6 3 2 1 # . (7) Let M be the set whose only element is the sun. Define addition and scalar multiplication on M by sun + sun = sun and c (sun) = sun. Is M a vector space under these operations? (8) Show that the set of vectors of the form ( a,b,c ) , where a,b,c are real numbers and b = a + c, is a subspace of R 3 . Find a basis for and di mension of this subspace....
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This note was uploaded on 04/23/2008 for the course MATH 20F taught by Professor Buss during the Winter '03 term at UCSD.
 Winter '03
 BUSS
 Linear Algebra, Algebra

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