Math54MT1(Teleman)

Math54MT1(Teleman) - A-1 = 1 det A 4 Consider the set W = p...

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Practice Midterm 1 February 26th, 2007 GSIs: Michaeel Kazi, Ivan Ventura 1. Let A = 1 2 - 3 2 - 1 1 2 - 3 2 - 1 0 1 1 . Find Col( A ), Nul( A ), Col( A 2 ), Nul( A 2 ). 2. Let B = 1 1 1 - 1 and C = 2 3 1 2 . Find the change of basis matrix P C→B . 3. For the following determine if the statement is true or false and justify your answer. (a) If A and B are invertible, then so is A + B . (b) If AB is invertible then so is A . (c) If T : m n is onto and B = { b 1 , . . . , b m } is a basis for m then T ( b 1 ) , . . . , T ( b m ) is a basis for
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Unformatted text preview: A-1 ) = 1 det( A ) 4. Consider the set W = { p ( x ) ∈ P 3 | p ( x ) = p (-x ) } (a) Prove, using the definition of a subspace, that W is a subspace of P 3 . (b) Find a linear transformation T such that W = ker( T ) [Hint: note that if p ( x ) = p (-x ) then p ( x )-p (-x ) = 0]. (c) Find the matrix of T . (d) Find a basis for W using your answer to 4c. 1...
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