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340spring06exam2sol

# 340spring06exam2sol - TOTAL SCORE(90 points possible MATH...

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TOTAL SCORE (90 points possible): MATH 340; EXAM # 2, April 11, 2006 (R.A.Brualdi) Discussion Section ( circle one ): Mon 8:50 Mon 12:05 Wed 8:50 Wed 12:05 NAME: 1. (9 points) Let T : R 3 R 3 be a linear transformation such that T (1 , 0 , 0) = (1 , 2 , 3) , T (0 , 1 , 0) = (2 , 1 , 4) , T (0 , 0 , 1) = (3 , 5 , 6) . Calculate T (4 , - 1 , 2). (4 , - 1 , 2) = 4(1 , 0 , 0) - 1(0 , 1 , 0) + 2(0 , 0 , 1) so that T (4 , - 1 , 2) = 4 T (1 , 0 , 0) - 1 T (0 , 1 , 0)+2 T (0 , 0 , 1) = 4(1 , 2 , 3) - 1(2 , 1 , 4)+2(3 , 5 , 6) = (8 , 17 , 20) . 2. [8 points] For each of the following pairs A, B of matrices, determine whether or not A and B are similar . Justify your answer in each case. (a) A = bracketleftBigg 1 2 2 3 bracketrightBigg , B = bracketleftBigg 2 2 1 2 bracketrightBigg . Yes No. Why? NO: they have different determinants (b) A = bracketleftBigg 3 - 1 1 3 bracketrightBigg B = bracketleftBigg 6 - 4 1 1 bracketrightBigg . Yes No. Why? No: they have different traces 3. (10 points) Let V and W be vector spaces of the same dimension n . Let v 1 , v 2 , . . . , v n be a basis of V . Let T : V W be a bijective linear transformation (isomorphism).

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340spring06exam2sol - TOTAL SCORE(90 points possible MATH...

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