notes6-4 - this to make arithmetic easier EXAMPLE Find an...

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SECTION 6.4 HOW TO GET AN ORTHOGONAL BASIS THE BASIC PROBLEM. Given a subspace W , we want to find an orthogonal basis for W . We can suppose we already have a basis for W , either because that’s how we got W in the first place, or if not, then we can produce a basis by starting with a nonzero vector in W , then adding vectors from W that aren’t in the span of the vectors we already have. So now we have a basis { u 1 , u 2 , u 3 , u 4 } for a subspace W . (Of course, 4 is not special!) Put u 1 in our proposed orthogonal basis B . The next vector in B must be orthogonal to u 1 and should surely come from u 2 . We know how to get such a vector: project u 2 onto u 1 and subtract the result from u 2 . Write this down and check out the spans. Do this again using the projection of u 3 onto the span of the first two. Write this down. CONTINUE! This is called the Gram-Schmidt Process. Since multiplying vectors by nonzero scalars does not change orthogonality or span or linear independence, we can do
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Unformatted text preview: this to make arithmetic easier. EXAMPLE. Find an orthogonal basis for the column space of A = -1 6 6 3-8 3 1-2 6 1-4-3 . What would happen if, say, the third column of A were a linear combination of the first two columns? Suppose A is a p × r matrix with linearly independent columns. Gram-Schmidt the columns of A , in order, and then normalize the columns, that is, divide each column by its length, to produce the columns of a matrix Q . What can we say about Q T Q ? Can we say that Q is invertible? Explain how we know there must be a matrix R such that QR = A . What can we say about R ? R = Q T A is an invertible upper triangular matrix. What we have here is called the QR factorization of A . It is heavily used in numerical methods for finding eigenvalues (p. 318) and solving equations (p. 414). HOMEWORK: SECTION 6.4...
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This note was uploaded on 03/06/2010 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

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notes6-4 - this to make arithmetic easier EXAMPLE Find an...

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