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MIT18_06S10_L04 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 4 Are we ready? Okay, ready for me to start. Ready for the taping to start in a minute. He's going to raise his hand and signal when I'm on. Just a minute, though, let them settle. Okay, guys. Okay, give me the signal, then, when you want me to start. Okay, this is linear algebra, lecture four. And, the first thing I have to do is something that was on the list for last time, but here it is now. What's the inverse of a product? If I multiply two matrices together and I know their inverses, how do I get the inverse of A times B? So I know what inverses mean for a single matrix A and for a matrix B. What matrix do I multiply by to get the identity if I have A B? Okay, that'll be simple but so basic. Then I'm going to use that to -- I will have a product of matrices and the product that we'll meet will be these elimination matrices and the net result of today's lectures is the big formula for elimination, so the net result of today's lecture is this great way to look at Gaussian elimination. We know that we get from A to U by elimination. We know the steps -- but now we get the right way to look at it, A equals L U. So that's the high point for today. Okay. Can I take the easy part, the first step first? So, suppose A is invertible -- and of course it's going to be a big question, when is the matrix invertible? But let's say A is invertible and B is invertible, then what matrix gives me the inverse of A B? So that's the direct question. What's the inverse of A B? Do I multiply those separate inverses? Yes. I multiply the two matrices A inverse and B inverse, but what order do I multiply? In reverse order. And you see why. So the right thing to put here is B inverse A inverse. That's the inverse I'm after. We can just check that A B times that matrix gives the identity. Okay. So why -- once again, it's this fact that I can move parentheses around. I can just erase them all and do the multiplications any way I want to.
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So what's the right multiplication to do first? B times B inverse. This product here I is the identity. Then A times the identity is the identity and then finally A times A inverse gives the identity. So forgive the dumb example in the book. Why do you, do the inverse things in reverse order? It's just like -- you take off your shoes, you take off your socks, then the good way to invert that process is socks back on first, then shoes. Sorry, okay. I'm sorry that's on the tape. And, of course, on the other side we should really just check -- on the other side I have B inverse, A inverse. That does multiply A B, and this time it's these guys that give the identity, squeeze down, they give the identity, we're in shape. Okay. So there's the inverse. Good. While we're at it, let me do a transpose, because the next lecture has got a lot to -- involves transposes.
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