MIT18_06S10_L17

MIT18_06S10_L17 - 18.06 Linear Algebra, Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 17 OK, here's the last lecture in the chapter on orthogonality. So we met orthogonal vectors, two vectors, we met orthogonal subspaces, like the row space and null space. Now today we meet an orthogonal basis, and an orthogonal matrix. So we really -- this chapter cleans up orthogonality. And really I want -- I should use the word orthonormal. Orthogonal is -- so my vectors are q1,q2 up to qn -- I use the letter "q", here, to remind me, I'm talking about orthogonal things, not just any vectors, but orthogonal ones. So what does that mean? That means that every q is orthogonal to every other q. It's a natural idea, to have a basis that's headed off at ninety-degree angles, the inner products are all zero. Of course if q is -- certainly qi is not orthogonal to itself. But there we'll make the best choice again, make it a unit vector. Then qi transpose qi is one, for a unit vector. The length squared is one. And that's what I would use the word normal. So for this part, normalized, unit length for this part. OK. So first part of the lecture is how does having an orthonormal basis make things nice? It certainly does. It makes all the calculations better, a whole lot of numerical linear algebra is built around working with orthonormal vectors, because they never get out of hand, they never overflow or underflow. And I'll put them into a matrix Q, and then the second part of the lecture will be suppose my basis, my columns of A are not orthonormal. How do I make them so? And the two names associated with that simple idea are Graham and Schmidt. So the first part is we've got a basis like this. Let's put those into the columns of a matrix. So a matrix Q that has -- I'll put these orthonormal vectors, q1 will be the first column, qn will be the n-th column. And I want to say, I want to write this property, qi transpose qj being zero, I want to put that in a matrix form. And just the right thing is to look at Q transpose Q. So this chapter has been looking at A transpose A. So it's natural to look at Q transpose Q. And the beauty is it comes out perfectly. Because Q transpose has these vectors in its rows, the first row is q1 transpose, the nth row is qn transpose. So that's Q transpose. And now I want to multiply by Q.
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That has q1 along to qn in the columns. That's Q. And what do I get? You really -- this is the first simplest most basic fact, that how do orthonormal vectors, orthonormal columns in a matrix, what happens if I compute Q transpose Q? Do you see it? If I take the first row times the first column, what do I get? A one. If I take the first row times the second column, what do I get? Zero. That's the orthogonality. The first row times the last column is zero.
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MIT18_06S10_L17 - 18.06 Linear Algebra, Spring 2010...

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