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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.
- Spring '08