test3-sp94

# test3-sp94 - C , then A is similar to C . (c) Prove or...

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MAS 4105 Test 3 February 25, 1994 Show all your work on the paper provided. Your work should be written in a proper and coherent manner. 50 points total. 1. (12 points) (a) Deﬁne linear transformation . (b) Let L : R 2 × 2 R 2 × 2 be deﬁned by L ( A ) = 3 A T . Determine whether or not L is a linear transformation. Prove your answer . (c) Let L : R 2 × 2 R 2 × 2 be deﬁned by L ( A ) = A - I . Determine whether or not L is a linear transformation. Prove your answer . 2. (10 points) Let L : V W be a linear transformation. (a) Deﬁne one-to-one transformation. (b) Prove that L is one-to-one if and only if the kernel of L consists only of the zero vector. 3. (13 points) Let A , B , and C be n × n matrices. (a) Deﬁne what it means to say that B is similar to A . (b) Prove that if A is similar to B , and B is similar to
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Unformatted text preview: C , then A is similar to C . (c) Prove or disprove: If A is similar to B , then det( A ) = det( B ). (d) Prove or disprove: If det( A ) = det( B ), then A is similar to B . 4. (15 points) Suppose that the matrix of a linear transformation L : R 3 R 3 relative to the standard basis is A = -2 9 6 1 0 3 4 Let w 1 = 1 , w 2 = 1 1-1 , and w 3 = 1 1 . (a) Find the matrix B representing L relative to the ordered basis F = [ w 1 , w 2 , w 3 ]. (b) Find a matrix S such that B = S-1 AS . (c) Let v = w 1-w 2 . Compute L ( v ) two ways, once using B and once using A . 1...
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## This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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