exam2-solution

# exam2-solution - Math 304 Examination 2 Linear Algebra...

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Unformatted text preview: Math 304 Examination 2 Linear Algebra Summer 2007 Write your name : Answer Key In problems 1–5 , 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 6–9 , 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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exam2-solution - Math 304 Examination 2 Linear Algebra...

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