# hw4 - F 9 Problem D Consider the linear map T R 11 → R 2...

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Winter 2011 Math 67 Linear Algebra Homework 4 Problems 3.3, 3.4, 3.5, 3.7 and 3.9 from Axler. Problem A. Let A and B be sets and f : A B a function. Prove that C f - 1 ( f ( C )) and that D = f ( f - 1 ( D )). Given an example where C 6 = f - 1 ( f ( C )). The assertion that D = f ( f - 1 ( D )) is wrong! You should only prove one containment: f ( f - 1 ( D )) D . Recall that for C A the image of C is f ( C ) = { f ( c ) : c C } , and that for D B the inverse image of D is f - 1 ( D ) = { c : f ( c ) D } . Problem B. Consider the subspace U of R 3 spanned by the vectors (1 , 2 , 3) t , (4 , 5 , 6) t , (2 , 1 , 0) t . Here (1 , 2 , 3) t denotes the column vector with entries 1 , 2 , 3 when read top to bottom. Determine the dimension of U . Describe U as null( T ) for some linear operator T : R 3 R 3 . Problem C. Consider the 2-by-2 matrix F = ± 1 1 0 1 ² What is the (1 , 2)-entry of
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Unformatted text preview: F 9 ? Problem D. Consider the linear map T : R 11 → R 2 that is described by the matrix ± 1-8 4 12 2-2 0 2 5 6 9 8 76 54 32 1 23 4 56 789 ² Let e 1 ,...,e 11 be the standard basis vectors of R 11 . Determine the image of 3 e 1 + e 9-e 3 under T . Is T a one-to-one map? (This goes without saying: give a reason why.) Is T surjective? (Dito.) Problem E. Determine the ranks of the linear maps that are described by the following matrices. 1 2 3 4 5 6 7 8 9 10 11 12 1 5 9 2 6 10 3 7 11 4 8-12 1...
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