Prelim2_summer2006 - Math 294 Summer Course Exam 2 1(10pts Find bases for the kernel and image of A where 3-9-4-6 1-3 A =-1 3 1 1 rref(A = 0 0-2 6

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Unformatted text preview: Math 294 Summer Course - Exam 2 07-11-2006 1) (10pts) Find bases for the kernel and image of A, where 3 -9 -4 -6 1 -3 A = -1 3 1 1 rref(A) = 0 0 -2 6 4 8 0 0 0 2 1 3 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 u, v, w are linearly dependent, then vector w must be a linear combination of u and v. 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) = a. Show T is linear. b. Show T is not an isomorphism. c. Find bases for the kernel and image of T . d. What is the rank of T ? e. State the Rank-Nullity Theorem and show it is satisfied by T . 0 f(t) dt. 4) (10pts) Which of the following sets are linear subspaces: a. S subset of R2 such that S = {(x, y) | y = x or y = -x} b. S subset of R3 such that S = {(x, y, z) | x + y + z = -3} a b a+d =0 c. S subset of R22 such that S = c d 5) (10pts) Consider the following two bases for P2 : U = {1, t, t2} B = {1, t - 1, (t - 1)2 } Find the change of bases matrices SBU and SU B . 1 ...
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This test prep was uploaded on 02/18/2008 for the course MATH 2940 taught by Professor Hui during the Spring '05 term at Cornell University (Engineering School).

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