notes6-4

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online