MIT18_06S10_L14

MIT18_06S10_L14 - 18.06 Linear Algebra, Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 14 OK. cameras are rolling. This is lecture fourteen, starting a new chapter. Chapter about orthogonality. What it means for vectors to be orthogonal. What it means for subspaces to be orthogonal. What it means for bases to be orthogonal. So ninety degrees, this is a ninety-degree chapter. So what does it mean -- let me jump to subspaces. Because I've drawn here the big picture. This is the 18.06 picture here. And hold it down, guys. So this is the picture and we know a lot about that picture already. We know the dimension of every subspace. We know that these dimensions are r and n-r. We know that these dimensions are r and m-r. What I want to show now is what this figure is saying, that the angle -- the figure is just my attempt to draw what I'm now going to say, that the angle between these subspaces is ninety degrees. And the angle between these subspaces is ninety degrees. Now I have to say what does that mean? What does it mean for subspaces to be orthogonal? But I hope you appreciate the beauty of this picture, that that those subspaces are going to come out to be orthogonal. Those two and also those two. So that's like one point, one important point to step forward in understanding those subspaces. We knew what each subspace was like, we could compute bases for them. Now we know more. Or we will in a few minutes. OK. I have to say first of all what does it mean for two vectors to be orthogonal? So let me start with that. Orthogonal vectors. The word orthogonal is -- is just another word for perpendicular. It means that in n-dimensional space the angle between those vectors is ninety degrees. It means that they form a right triangle. It even means that the going way back to the Greeks that this angle that this triangle a vector x, a vector x, and a
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vector x+y -- of course that'll be the hypotenuse, so what was it the Greeks figured out and it's neat. It's the fact that the -- so these are orthogonal, this is a right angle, if -- so let me put the great name down, Pythagoras, I'm looking for -- what I looking for? I'm looking for the condition if you give me two vectors, how do I know if they're orthogonal? How can I tell two perpendicular vectors? And actually you probably know the answer. Let me write the answer down. Orthogonal vectors, what's the test for orthogonality? I take the dot product which I tend to write as x transpose y, because that's a row times a column, and that matrix multiplication just gives me the right thing, that x1y1+x2y2 and so on, so these vectors are orthogonal if this result x transpose y is zero. That's the test. OK. Can I connect that to other things? I mean -- it's just beautiful that here we have we're in n dimensions, we've got a couple of vectors, we want to know the angle between them, and the right thing to look at is the simplest thing that you could imagine, the dot product. OK. Now why? So I'm answering the question now why -- let's add some justification to
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This note was uploaded on 08/22/2011 for the course MATH 1806 taught by Professor Strang during the Fall '10 term at MIT.

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MIT18_06S10_L14 - 18.06 Linear Algebra, Spring 2010...

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