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Unformatted text preview: Math 304 Examination 1 Linear Algebra Summer 2007 Write your name : Answer Key (2 points). In problems 15 , circle the correct answer. (5 points each) 1. If A is a 3 3 matrix, then A is a singular matrix if and only if the linear system A x = is inconsistent. True False Solution. The statement is false, because the homogeneous system A x = 0 is always consistent (since x = 0 is a solution). 2. If A is an invertible 3 3 matrix, then ( A T )- 1 = ( A- 1 ) T . True False Solution. The statement is true, and here is one way to see why. Since A- 1 A = I , taking the transpose shows that A T ( A- 1 ) T = I . By the definition of inverse matrix, this equation means that ( A T )- 1 = ( A- 1 ) T . 3. If v 1 , v 2 , and v 3 are elements of a vector space V , then the span of v 1 , v 2 , and v 3 (that is, the set of all linear combinations of v 1 , v 2 , and v 3 ) is a subspace of V . True False Solution. True; this is one of the fundamental ways of creating sub- spaces. The statement is Theorem 3.2.1 on page 128 of the textbook. 4. A linear system A x = b is consistent if and only if the column vector b can be written as a linear combination of the column vectors of the matrix A . True False Solution. True, because the matrix product A x can be interpreted as a linear combination of the columns of the matrix. The statement is Theorem 1.3.1 on page 37 of the textbook....
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