final - Questions from 18.06 Final, Fall 2003 1. Suppose A...

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Questions from 18.06 Final, Fall 2003 1. Suppose A = LU where L = 1 0 0 2 1 0 - 2 3 1 , U = 5 0 5 1 0 3 3 0 0 0 0 0 . (a) What are the dimensions of the 4 fundamental subspaces associated with A ? (b) Give a basis for each of the 4 fundamental subspaces. 1
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N ( A ) R ( A ) C ( A ) N ( A T ) 2
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2. Let F be the subspace of R 4 given by F = { ( x, y, z, w ) : x - y + 2 z + 3 w = 0 } . Let P be the projection matrix for projecting onto F . (Many of the subquestions can be answered independently of the others.) (a) Give an orthonormal basis { v 1 , ··· , v k } for the orthogonal complement to F . (b) Find an orthonormal basis { w 1 , ··· , w l } for F . Explain how you proceed. 3
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P for projecting onto F . (c) What are the eigenvalues of P ? Give them with their multiplicities. (d) What is the projection of 1 1 1 1 onto F ? 4
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final - Questions from 18.06 Final, Fall 2003 1. Suppose A...

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