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Unformatted text preview: linear independence, we can do this to make arithmetic easier. EXAMPLE. Find an orthogonal basis for the column space of -1 6 6 3-8 3 1-2 6 1-4-3 . Use the vectors in the orthogonal basis weve found as columns of a matrix Q . Compute Q T Q and Q T A , divide the rows of Q T A by the corresponding entries in Q T Q to get a matrix R . Finally, calculate QR . This always works and is usually done by using vectors in an orthonormal basis, rather than just an orthogonal basis. When the matrix A has linearly independent columns, then using Gram-Schmidt we can nd a matrix Q whose columns are an orthonormal basis for Col A . If we put R = Q T A , then we get the QR factorization of A as A = QR , where R is an invertible upper triangular matrix. HOMEWORK: SECTION 6.4...
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- Spring '08