practice_exam3 - Math 3000, Fall 2011, Practice Sheet for...

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Unformatted text preview: Math 3000, Fall 2011, Practice Sheet for Exam 3 1. Let A = 1 3- 1 . Find A- 1 . 2. Suppose A and B are n n matrices. Prove that the following two statements are equivalent a. det ( A- B ) = 0 b. there exit nonzero vectors ~v such that A~v = B~v 3. Let V < R 4 be the two-dimensional subspace spanned by the vectors b 1 = 1 / 2 1 / 2 1 / 2 1 / 2 b 2 = 1 / 2- 1 / 2- 1 / 2 1 / 2 a. Show that B = ( b 1 ,b 2 ) is an orthonormal basis for V . b. Consider the linear transformation T : R 4 R 4 defined by projection T ( ~v ) = proj V ( ~v ). Write down a matrix representing T with respect to the standard basis of R 4 . c. Consider the linear transformation T : R 4 V defined by T ( ~v ) = proj V ( ~v ). Write a matrix repesenting T with respect to the standard basis on R 4 and the basis B on V . 4. Consider the matrix A = 3 4- 1 1 Find det ( A )....
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practice_exam3 - Math 3000, Fall 2011, Practice Sheet for...

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