3e-spring2011-exam_2_sample

3e-spring2011-exam_2_sample - Math 3E Exam #2 Sample Laney...

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Unformatted text preview: Math 3E Exam #2 Sample Laney College, Spring 2011 Fred Bourgoin 1. (10 points) Is the set W of all 2 2 matrices [ a b c d ] such that a + b + c + d = 0 a subspace of R 2 2 ? Prove or disprove. 2. (5 points) A basis for P 2 is B = { x 2 + 1 , x + 1 , x 2 + x } . If [ f ] B = 3 1 2 , what is f ? 3. (8 points) Consider the subset B = { x 2 + 1 , x + 1 , x 2 + x } of P 2 . (a) Does B span P 2 ? Justify. (b) Is B a basis for P 2 ? Justify. 4. (10 points) Consider the matrix A = 1 1 1 4 2 1 3 5 2 0 4 2 . (a) Find a basis for im( A ). (b) Find a basis for ker( A ). 5. (5 points) Let A be an n n matrix. Give five statements equivalent to A is invertible. 6. (5 points) Is the subset W = { ax 2 + bx + c P 2 | c = 2 } of P 2 a subspace of P 2 ? Prove or disprove. 7. (10 points) Let T : R 2 R 2 be the linear transformation defined by T ([ x y ]) = [ x + 2 y 2 x y ] , and consider the basis B = {[ 1 2 ] , [ 2 ]} for R 2 . Find the B-matrix of T . 8. (5 points) Do the following form a basis for R 2 2 ? Justify. [ 1 1 1 1 ] , [ 1 0 0 2 ] , [ 0 1 0 2 ] 9. (5 points) Find a basis the subspace of R 3 defined by W = span 1 2 2 , 3 2 1 , 11 10 7 , 7 6 4 . What is the dimension of W ? 1 10. (6 points) Let W be the subspace of P 4 of all polynomials of the form ax 4 + bx 2 + c . Show that the linear transformation T : W P 2 defined by T ( ax 4 + bx 2 + c ) = ax 2 + bx + c is an isomorphism. 11. (10 points) Let T : R 3 3 R be defined by T ( A ) = tr( A ). [Recall that tr( A ), the trace of A , is the sum of the diagonal entries of A .] (a) Is T is linear transformation? Prove or disprove. (b) Is T an isomorphism? Prove or disprove. 12. (10 points) Let T : R R be defined by T ( x ) = ax + b . (a) For what values of a and b is T a linear transformation? (b) For what values of a and b is T an isomorphism? 13. (10 points) True of False? (You do not need to justify your answers.) (a) The image of a 3 4 matrix is a subspace of R 4 . (b) If T is an isomorphism, then T 1 must be an isomorphism as well. (c) The column vectors of a 5 4 matrix must be linearly independent. (d) If A is a 5 6 matrix of rank 4, then the nullity of A is 1....
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3e-spring2011-exam_2_sample - Math 3E Exam #2 Sample Laney...

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