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Unformatted text preview: Math 304 Examination 2 Linear Algebra Summer 2007 Write your name : Answer Key In problems 15 , circle the correct answer. (5 points each) 1. If A is a 12 5 matrix (that is, A has 12 rows and 5 columns), then the null space of A has dimension at least 7. True False Solution. False. The null space is a subspace of R 5 , so its dimension might be anything between 0 and 5, but the dimension cannot possibly be as large as 7. What is true is that the dimension of the null space of A T is at least 7. 2. The function L : R 2 R 1 defined by L ( x ) = bardbl x bardbl (that is, the norm of x ) is a linear transformation. True False Solution. False. If x negationslash = , then bardbl x bardbl negationslash = bardbl x bardbl , so the function does not preserve scalar multiplication. Moreover, bardbl x + y bardbl is not always equal to bardbl x bardbl + bardbl y bardbl , so the operation does not preserve addition. 3. The matrix parenleftbigg 1 2 0 3 parenrightbigg is similar to the matrix parenleftbigg 2 4 0 6 parenrightbigg . True False Solution. False. These two matrices have different determinants, so the matrices cannot be similar matrices. 4. If A is a 3 3 matrix of rank 2, then the dimension of the null space of A T (the transpose) is equal to 2. True False Solution. False. Since A and A T have the same rank, the rank of A T is equal to 2, so by the rank-nullity theorem, the dimension of the null space of A T is equal to 1. 5. If a 2 2 matrix of real numbers has purely imaginary eigenvalues, then the determinant of the matrix is negative. True False Solution. False. The eigenvalues will be complex conjugates, say ib for some real number b , so the determinant, which is equal to the product of the eigenvalues, will be b 2 , which cannot be negative. In problems 69 , fill in the blanks. (7 points per problem) June 29, 2007 Page 1 of 5 Dr. Boas Math 304 Examination 2 Linear Algebra Summer 2007 6. If L is the linear operator on R 2 that doubles the length of each vector and also rotates each vector by 30 counterclockwise, then the standard...
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