prelim2_summer2006 - Math 294 Summer Course Exam 2 1(10pts...

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Math 294 Summer Course - Exam 2 07-11-2006 1) (10pts) Find bases for the kernel and image of A , where A = 3 - 9 - 4 - 6 - 1 3 1 1 - 2 6 4 8 rref ( A ) = 1 - 3 0 2 0 0 1 3 0 0 0 0 2) (10pts) T or F Are the following in general, true or false. If false state why! a. There exists a 3 × 3 matrix A such that im( A ) = ker( A ). b. If two nonzero vectors are linearly dependent, then each of them is a scalar multiple of the other. c. If vectoru,vectorv, vectorw are linearly dependent, then vector vectorw must be a linear combination of vectoru and vectorv . d. The column vectors of a 4 × 5 matrix must be linearly dependent. e. A change of basis is an isomorphism. f. Linear spaces are restricted to vectors and matrices. 3) (10pts) Consider the transformation T : V -→ V , where V = span { 1 , cos t, sin t } and T ( f ) = integraltext π 0 f ( t ) dt . a. Show T is linear.
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