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MidTermSample

# MidTermSample - x = 1 1 1(c Find the projection of the...

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APAM-Linear Algebra: Midterm exam sample 2009

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1. Let A be a 4 by 3 matrix with linearly independent columns. (a) What are the dimensions of the four fundamental subspaces C ( A ) ,N ( A ) ,C ( A T ) ,N ( A T ) ? (b) Describe explicitly the nullspace N ( A ) and the row space C ( A T ) of A . (c) Suppose that B is a 4 by 3 matrix such that the matrices A and B have exactly the same column spaces C ( A ) = C ( B ) and the same nullspaces N ( A ) = N ( B ) . Are you sure that in this case A = B ? YES NO Prove that A = B or give a counterexample where A 6 = B . 2. (a) Compute the projection of the vector b = 1 1 1 onto the column space of A = 1 0 2 0 1 5 - 1 0 - 2 . (Hint: First check whether A has linearly independent columns.) (b) Find the least-square solution ˆ x for the system 1 0 0 1 - 1 0
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Unformatted text preview: x = 1 1 1 . (c) Find the projection of the vector 1 1 1 onto the column space of A = 10000 0 2 1 5-1-2 . (Hint: No computations!) 3. Consider the matrix A = 1 1 1 1 1 2 3 5 1 3 5 9 . (a) What is the rank of A ? 1 (b) Find a matrix B such that the column space C ( A ) of A equals the nullspace N ( B ) of B . (c) Which of the following vectors belong(s) to the column space C ( A ) : 1-2 1 , 2-2 , 2 4 8 , 1-1 1-1 ? 2...
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MidTermSample - x = 1 1 1(c Find the projection of the...

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