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# exam1 - Math 304 Examination 1 Linear Algebra Summer 2006...

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Unformatted text preview: Math 304 Examination 1 Linear Algebra Summer 2006 Write your name : Answer Key (2 points). In problems 1–5 , circle the correct answer. (5 points each) 1. Every linear system of three equations in four unknowns is consistent. True False Solution. False. In a consistent system A x = b , the vector b has to be a linear combination of the columns of the matrix A . An example of an inconsistent system of three equations in four unknowns is x 1 + x 2 + x 3 + x 4 = 1 x 1 + x 2 + x 3 + x 4 = 2 x 1 + x 2 + x 3 + x 4 = 3 . 2. Three vectors in R 2 are always linearly dependent. True False Solution. True. This is a particular case of Theorem 3.4.1 on page 147. Three vectors in a two-dimensional vector space are linearly dependent. 3. If A is a square matrix such that det( A ) = 0, then the homogeneous system A x = has infinitely many solutions. True False Solution. True. This follows from (for example) Theorem 1.4.2 on page 65 or the subsequent Corollary 1.4.3. 4. If v 1 , v 2 , and v 3 are elements of a vector space V , then the span of the vectors v 1 , v 2 , and v 3 is a subspace of V . True False Solution. True. This is the statement of Theorem 3.2.1 on page 128. 5. If A is a square matrix that is singular, then the matrix A T (that is, the transpose of A ) is singular too. True False Solution. True. If A is singular, then det( A ) = 0. But det( A ) = det( A T ) (Theorem 2.1.2 on page 96), so det( A T ) = 0, and hence A T is singular. June 9, 2006 Page 1 of 4 Dr. Boas Math 304 Examination 1 Linear Algebra Summer 2006 In problems 6–9 , fill in the blanks. (7 points per problem) 6. parenleftbigg 3 5 1 − 2 0 2 parenrightbigg 2 1 1 3 4 1 = parenleftbigg 15 19 4 parenrightbigg 7. det 2 0 1 0 1 1 6 2 1 1 8 3 = 18 8. If 1 0 0 0...
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exam1 - Math 304 Examination 1 Linear Algebra Summer 2006...

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