WEBVTT
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I've been talking, I've been multiplying matrices already, but certainly it's time for
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me to discuss the rules for matrix multiplication and the interesting part is that many ways you
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can do it and they all give the same answer. And they're all important. So matrix multiplication
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and then come inverses. So we mentioned the inverse of a matrix, but that's a big
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deal. Lots to do about inverses and how to find them. Okay, so I'll begin with how to
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multiply two matrices. First way. Okay, so suppose I have a matrix A multiplying a matrix
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B and giving me a result. Well, I could call it C A times B. Okay, so let me just review
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the rule for this entry. That's the entry in row I and column J. So that's the IJ entry.
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Right there is C IJ. We always write the row number and then the column number. So I might
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maybe I take it C three four just to make it specific. So instead of IJ, let me use
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numbers C three four. So where does that come from? The three four entry. It comes from row
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three here, row three and column four. As you know, column four. And can I just write down
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or can we write down the four millifuert? C three four is if we look at the whole row
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and the whole column, the quick way for me to say it is row three of A. I could use a dot
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for dot product. I won't often use that actually. Dot column four of B. And but this gives
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us a chance to just like use a little matrix notation. What are the entries? What's this
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first entry in row three? The first, the first that number that's sitting right there
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is A. So it's got two indices and what are they? Three one. So there's an A three one there.
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Now what's the first guy at the top of column four? So what's sitting up there? B one
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four. Right. So that this dot product starts with A three one times B one four. And then
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what's the next? So this is like I'm accumulating this sum. Then comes the next guy A three two.
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And columns times B two four second row. So it's B A three two B two four and so on. Just
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practice with indices. Oh, let me even practice with a summation formula. So this is I most
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of the course I use whole vector a very seldom get down to the details of these particular
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entries. But here we better do it. So I'm at some kind of a sum right of things in row
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three column K should I say times things in row K column four. You see that that's what
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we're seeing here. This is K is one. Here K is two on the long so up so the sum goes all
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the way along the row and down the columns say one to end. So that's what the A the C three
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four entry looks like a sum of A three K B K four. This takes a little practice to do that.
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Okay. And oh well maybe I should say when are we allowed to multiply these matrices?
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What are the shapes of these things? The shapes are if we allow them to be not necessarily
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square matrices. If they're square they've got to be the same size. If they're rectangular
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they're not the same size. If they're rectangular this might be well I always think of A as
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M by N M rows and columns. So that sum goes to N. Now what's the point how many rows
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does B have to have? N. N. The number of rows in B the number of guys that we meet coming
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down has to match the number of ones across. So B will have to be N by something whatever
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P. So the number of columns here has to match the number of rows there and then what's the
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result? What's the shape of the result? What's the shape of C the output? Well it's got
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the same N rows or it's got N rows and how many columns? P. N by P. Okay. So there are
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N times P's little numbers in there entries and each one looks like that. Okay. So that's
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the standard rule. That's the way people think of multiplying matrices. I do it too.
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But that's I want to talk about other ways to look at that same calculation. Looking
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at whole columns and whole rows. Okay. So can I do A, B, C again? A, B equaling C again.
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But now tell me about, yeah let me I'll put it up here. So here goes A again times B
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reducing C. And again this is N by N. This is N by P and this is N by P. Okay. Now I want
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to look at whole columns. I want to look at the columns of in fact here's the second
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way to multiply matrices. Because I'm going to build on what I know already. How do I multiply
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a matrix by a column? How do I I know how to multiply this matrix by that column? So I
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call that column one. That tells me column one of the answer. The matrix times the first
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column is that first column. Because none of this stuff enters that part of the answer.
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The matrix times the second column is the second column of the answer. You see what I'm
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saying? That I could think of multiplying a matrix by a vector which I already knew
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how to do. And I can think of the vector I can think of just P columns sitting side by
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side just like resting next to itself. And I multiply A times B is one of those and I get
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the P columns of the answer. Do you see this is slight nice. To be able to think okay.
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Matrix multiplication works so that I can just think of having several columns multiplying
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by A and getting the columns of the answer. So like here's column one. I call that column
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one. And what's going in there in A times column one. So that's the picture a column of
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the time. So what does that tell me? What does that tell me about these columns? These columns
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are combinations because we've seen that before of columns of A. Every one of these comes
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from A times this. And A times a vector is a combination of the columns of A. And it
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makes sense because the columns of A have length A and the columns of C have length A. And
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every column of C is a some combination of the columns of A and it's these numbers in
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here that tell me what combination it is. Do you see that? That's that out if that
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isn't that answer C I'm seeing stuff that's called that's combinations of these columns.
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Now suppose I look at it that's two ways now. The third way is look at it by rows. So
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now let me change the rows. So now I can think of a row of A multiplying all these rows
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here and producing a row of the plus. So this row takes the combination of these rows
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and that's the answer. So these rows of C are combinations of what? Only route of
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finish that. The rows of C when I have a matrix B it's got rows and I multiply by A. And
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what does that do? It mixes the rows up. It creates combinations of the rows of B. Thanks.
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Rows of B. That's what I wanted to see. That this answer I can see where the pieces are
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coming from. The rows in the answer are coming as combinations of these rows. The columns
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in the answer are coming as combinations of those columns. And now that's three ways.
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Now you can say okay what's the fourth way? The fourth way. Now we've got like the regular
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way, the column way, the row way and what's left. The one that I can I want to tell you
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about what one way is columns times rows. What happens if I multiply? So this was row
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times column. It gave a number. Now I want to ask you about column times row. What does
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if I multiply a column of A times row of B? What's the shape of my ending upwards? So if
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I take a column times row, that's definitely different from taking a row times a column.
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So a column of A was what's the shape of a column of A? M by what? A column of A is a column.
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It's got M entries and one column. And what's a row of B? It's got one row and P column.
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So what's the shape? What do I get if I multiply a column by a row? I get a big name term.
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I get a full size name term. If I multiply a column by a row, I get, should we just do
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one? Let me take the column 2, 3, 4, times the row 1, 6. That is a row.
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That is a product there. I mean, just following the rules of matrix multiplication, those
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rules are just looking like kind of petite kind of small because the rows here are so short
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and the columns there are so short, but they're the same length. So what's the answer?
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What's the answer if I do 2, 3, 4, times 1, 6? Just for practice. Well, what's the first row of the
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answer? 2, 12. And the second row of the answer is 3, 18. And the third row of the answer
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is 4, 24. Actually, what am I, I mean, that's a very special matrix there. Very special matrix.
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What can you tell me about its column, the columns of that matrix? They're multiples of this
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guys. Right? They're multiples of that one. Which columns are rules? We said that the columns
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of the answer were combinations, but there's only, you could take a combination of one
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guy, it's just a multiple. The rows of the answer, what can you tell me about those three rows?
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They're all multiples of this row. They're all multiples of one, six, as we expect.
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But I'm getting a full size matrix. And now, just a completed thought, if I have,
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now let me write down the fourth way. A, B is a thumb of columns of A times rows of B.
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So, for example, if my matrix was 2, 3, 4 and then had another column, say, 7, 8, 9,
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and my matrix here has, they started with one six, and then had another column like 0, 0.
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Then, here's the fourth way. Okay? I've got two columns there, I've got two rows there.
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So, the beautiful rule is, seeing the whole thing by columns and rows is that I can take
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the first column times the first row and add the second column times the second row.
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So, that's the fourth way that I can take columns times rows, first column times
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the first row, second column times the second row, and add. Actually, what will I get?
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What will the answer be for that matrix multiplication? Well, this one is just going to give
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us 0. So, in fact, I'm back to this, that's the answer for that matrix multiplication.
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I'm sort of like happy to put up here these facts about matrix multiplication,
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because it gives me a chance that you write down special matrices like this.
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This is a special matrix. All those rows lie on the same line.
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All those rows lie on the line through one six. If I draw a picture of all these row vectors,
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they're all the same direction. If I draw a picture of these two column vectors,
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they're in the same direction. Later, I would use this language.
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Not too much later, either. I would say the row space, which is like all the combinations
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of the rows, is just a line for this matrix. The row space is the line through the
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vector one six. All the rows lie on that line. And the column space is also a line. All the columns lie on the line through the vector two, three, four.
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So, this is like a really minimal matrix, and it's because of these ones.
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Okay, so that's the third. Now, even when you take, I want to say one more thing about matrix multiplication,
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while we're on the subject. And it's this. You could also multiply,
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you could also cut the matrix into block, and do the multiplication by block.
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Yes, that's actually so useful that I want to mention it.
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Block, multiply. So, I can take my matrix A, and I can chop it up, like maybe just for simplicity,
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let me chop it into four square blocks, the both of square, let's just take a nice table.
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And these, suppose this square also, same size. So, these sizes don't have to be the same.
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What they have to do with match property. Here they certainly will match.
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So, here's the rule for block multiplication. That if this has blocks like A,
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so maybe A1, A2, A3, A4, or the block here, and these blocks are B1, B2, B3, and B4,
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then the answer, I can find block. I can find that block. And if you tell me what's in that block,
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then I'm going to be quiet about matrix multiplication for the rest of the day.
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What goes into that block? You see these might be, this matrix might, these matrices might be by 20 by 20,
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with blocks that are 10 by 10 to take the easy case for all the blocks to the same shape.
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And the point is that I could multiply those by block. And what goes in here?
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What's that block in the answer? A1, B1.
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As a matrix, kind of a matrix, it's the right size, 10 by 10, and any more?
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Plus, what else goes in here? A2, B3. Right? It's just like block row times block column.
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I don't, nobody I think, not even GALS could see instantly that it works.
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But somehow, if we check it through, all five ways we're doing the same multiplications.
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So this familiar multiplication is what we're really doing when we do it by columns, by rows, by columns, times rows, and by block.
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Okay, I just have to like get the rules for matrix multiplication.
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Okay.
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All right, I'm ready for the second topic, which is input.
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Okay, ready for input.
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And let me do it for square matrices first.
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Okay, so I've got a square matrix A.
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And it may or may not have an inverse, right?
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Not all matrices have inverses. In fact, that's the most important question.
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You can ask about the matrix if it's square, if you know it's square, is it invertible or not?
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If it is invertible, then there is some other matrix, so I call it A inverse.
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And what's the, what's if A inverse exists?
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So this is if, there's a big F here. If this matrix exists, and it'll be really central, let's be drawn, when does it exist?
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And then if it does exist, how would you find it?
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But what's the, what's the, the equation here that I have, that I have to finish now?
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This matrix entities this, multiply as A and produce it.
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I think the identity. And actually there's a little more to it.
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Because normally, I mean that's, right now that's like a what inverse, sitting on the left of A.
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But a real, an inverse, it's for a square matrix, could be on the right.
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So, so this is true too.
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That if I have it, in fact this is not, this is probably the, this is something that's not easy to prove, but it's worth.
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That a left is that for square matrices, a left inverse is also a right inverse.
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If I can find the matrix on the left, that gets the identities, and also that matrix on the right, will produce that, I guess.
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For rectangular matrices, we'll see a left inverse, that isn't a right one.
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In fact, the shapes wouldn't allow. But for square matrices, the shapes allow it, and it happens.
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If A has an inverse. Okay. So, give me some cases.
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See, I hate to be negative here, but let's talk about the case with no examples.
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So, so this is, this is, these matrices are called invertible, or non-singular.
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So, this is a good one. And we want to be able to identify how this would give a matrix, has it got an inverse.
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Can I talk about the singular case? No inverse.
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Alright. Best to start with an example. Tell me an example. Let's just get an example up here.
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Let's make it 2x2, of a matrix that has not got an inverse. And let's see what.
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Let me, let me write one up. No inverse. Let's, let's see what.
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Let me write out one, three, two, three.
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Why does that matrix have no inverse? You can, you can answer that various way.
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Give me one reason. Well, if you know about determinants, what you're not supposed to be.
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You could take this determinant and you would get zero. Okay.
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Now, alright. Let me, let me ask you other reason. And as for other reasons that that matrix isn't invertible.
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I can use, use what I'm saying here. Suppose, suppose A times some other matrix gave the identity.
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Why is that not possible? Because, oh yeah, I'm thinking about columns here.
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If I multiply this matrix A by some other matrix, then the result, what can you tell me about the columns?
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They're all, multiple of those columns, right? If I multiply A by another matrix,
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that the product has columns that come from those columns. So, can I get the identity matrix? No way.
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The columns of the identity matrix like one zero, it's not a combination of those columns.
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Because those two columns lie on both lines on the same line. Every combination is just going to be on that line and I can't get one zero.
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So, you see, you see that, that sort of column picture of the matrix not being invertible.
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In fact, here's another reason. This is even more important. Well, how can I say more important?
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All of them. This is another way to see it. A matrix has no inverse. Yeah, it hits out. This is important.
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A matrix has no, a square matrix won't have an inverse if there's, if, no inverse because I can solve, I can find an X
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of, a vector X with A times, this A times X giving zero. That's the reason I like that.
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That matrix won't have an inverse tenu, this is the, well, let me change the I the U. So, tell me a vector X that calls AX equals zero.
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I mean, this is like the key equation in mathematical, the key equation that's zero on the right hand side. So, what's the X?
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So, tell me an X here. So, now I'm going to put slip in the X that you tell me and I'm going to get zero.
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What X would do that job? Three and negative one? Is that the one you picked or, yeah, or another one? Well, if you've had zero and zero, I'm not so excited.
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That would always work. So, it's really the fact that this vector isn't zero. It's important. It's a non-zero vector and three negative one would do it.
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That just says three of this column minus one of that column is zero column. Okay.
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So, now I know that A couldn't be inverted. What's the, what's the reasoning? If, if AX is zero, suppose I multiply by A inverse.
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Yes. Oh, here's the reason. Here, here's, this is why this spells disaster for an inverse.
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A matrix can't have an inverse if some combination of the columns gives us. It gives nothing. Because I could take A and C equals zero.
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I could multiply by A inverse. And what would I discover? Suppose I take that equation and I multiply by, if A inverse existed, which of course I'm going to come as a conclusion is 10.
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Because if it existed, if it wasn't a inverse for this, it would still be matrix. I would multiply that equation by that inverse and I would discover X is zero.
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If I multiply 8 by A inverse, on the left I get X. If I multiply by A inverse, on the right I get zero. So, I would discover X was zero. But, X is not zero.
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That this guy wasn't here. There it is. Three minus one. So, conclusions. Only take us sometimes a really work of that conclusion.
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Our conclusion will be that, that, that's non-invertible matrices. Think of a matrices. Some combinations of, some combinations and their columns gives us zero points.
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They take some vector X into zero. And there's no way A inverse can recover. Right? That's what this equation says. This equation says I take this vector X and multiply by A into zero.
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But then, when I multiply by A inverse, I can never escape from zero. So, there couldn't be an A inverse. Where here? Okay. Now fix, all right. Now, let me take a, all right. Back to the positive side.
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Let's take a matrix that does that A inverse. And, why not invert it? Okay. So, let me take on the third board a matrix. So, I fixed that up a little.
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Let me make a matrix that has seven entries. Well, let me say one, three, two, what shall I put there? Well, don't put six, I guess you. Right?
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Do you want any, any favorites here? One? Eight? I don't care. Seven? Seven. Okay. Seven is a lucky. All right. Okay. Okay. So, now what's our idea? We believe that this matrix is invert. Those who like determinants have quickly taken its determinant and found it was zero.
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Those who like columns and found that that department is not totally popular yet. But those who like columns will look at those two columns and say, hey, they point in different directions.
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So, I can get anything. Now, let me see. What do I mean? How am I going to compute A inverse? So, A inverse, here's A inverse. And I have to find. And what do I get when I do this multiplication? The identity.
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You know, forgive me for taking two by two, but it's good to keep the conversation manageable and let the ideas come out. Okay. Now, what's the idea I want? I'm looking for this matrix A inverse. Right now, it's a good for numbers to find.
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Let me look at the first column. Let me take this first column, A, B. What's up there? What a quacy. Tell me this. What equation does the first column satisfy? The first column satisfies A times that column is one zero.
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The second column, B, D, satisfies A times that second column is zero one. You see that finding the inverse is like solving two systems. One system when the right hand side is one here. I'm just going to split it into two pieces.
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I don't even need to rewrite it. I can say to take A times. So, let me put it here. A times column J of A inverse is column J of the identity.
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I've got in equation. I've got two in this. And they have the same matrix A, but they have different right hand sides. The right hand sides are just the columns of the identity. This guy and this guy.
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And these are the two solutions. Do you see what I'm looking at that equation by columns? I'm looking at A times this column giving that guy and A times that column giving that guy. So, essentially, so this is like the gout. We're back to gout. We're back to solving systems of equations, but we're solving. We've got two right hand sides instead of one.
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That's where Jordan comes in. So, it's the very beginning of the lecture I mentioned. Gout Jordan, let me write it up again.
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Here's the gout Jordan idea. Gout Jordan is solved two equations at once.
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Let me show you how the mechanics show. How do I solve a single equation? So, the two equations are 1, 3, 2, 7. Multiply A, B gives 1, 0.
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And the other equation is the same 1, 3, 2, 7. Multiplying C, D gives 0, 1. Okay. That'll tell me the two columns of the inverse. I'll have the inverse. In other words, if I can solve with this matrix A, if I can solve with that right hand side and that right hand side, I'm in vertical.
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And Jordan sort of said to gout, solve them together. Look at the matrix. If we just solve this one, I would look at 1, 3, 2, 7. And how do I deal with the right hand side? I stick it on as an extra column.
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Remember that's this augmented matrix. That's the matrix when I'm watching the right hand side at the same time doing the same thing to the right side that I do on the left.
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So I just carry it along as an extra column. Now I'm going to carry along two extra columns.
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And I'm going to do whatever gout wants. Right. I'm going to do elimination. I'm going to get this to be simple. And this thing will turn into the inverse. This is what comes.
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I'm going to do elimination steps to make this into the identity. And lo and the whole the inverse will show up here.
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Let's do it. OK. So what are the elimination steps? So you see here's my matrix A. And here is the identity. Like stuck on augmented only.
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Yeah. Did I? Oh, no, they weren't. Sorry. Thanks. Thank you very much. And I've got them right. OK. Thanks. OK. So let's do elimination. All right. It's going to be simple. Right. So I take two of this row away from this row.
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So this row stays the same. And two of those come away from this. That leads me with a zero and a one and two of these away from this. Is that what that? Is that what you're getting after one elimination step?
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Let me sort of separate the left half from the right half. So two of that first row got subtracted from the second row.
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Now, now this is an upper triangular form. Goult was quick, but Jordan says to use elimination upwards subtract. I'm multiple of equation two from a crazy one to get rid of the three.
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So let's go the whole way. So now I'm going to this guy is fine. But I'm doing what do I do now? What's my final steps that produces the inverse? I multiply this by the right number to get up to there to remove that three.
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So I guess since this is a one, there's the pivot sitting there. I multiply it by three and subtract from that. So what do I get? I'll have one zero. Oh, yeah, that was my whole point. I multiply this by three and subtract from that, which will give me.
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Seven and I multiply this by three and subtract from that, which gives me a minus three.
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And what's my hope, believe here, here I started with with a and the identity and I ended up with the identity and who that better be a.
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That's the goult Jordan idea start with this long matrix double length a i eliminate eliminate until this part is down to I. And this one will must be for some reason we got to find the reason must be a.
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So I just checked that it works. Let me just check that can I multiply this matrix this part times a I'll carry a over here.
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And just do that multiplication I you'll say I'll do it the old fashioned way seven minus six is a one twenty one minus twenty one is a zero minus two plus two is a zero minus six plus seven is a one check.
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So that is the end that's the goult Jordan idea so you'll one of the homework problems or more than one for when the will ask you to go through those sets.
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I think you just got to go through goult Jordan a couple of times.
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But I try I like yeah just to see the mechanics but the important thing is why is like what happened why did we why do we get a and
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that let me ask you that we got so we take we do row reduction we do elimination on this long matrix a i until the first half is up.
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Then the second half is a inverse well how do I see that let me put up here how I see that so here's my here's my goult Jordan thing and I'm doing stuff to it.
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Well a whole lot of e remember those are those elimination matrix those are those are the things that we figured out last time yes that's what an elimination step is it's in matrix form I'm multiplying by some e and the results well so I'm multiplying by a whole bunch of e so I get a can I call the overall matrix e.
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That's the elimination matrix the product of all those little pieces what do I mean by little pieces well there was an elimination matrix that subtracted two of that away from that.
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Then there was elimination matrix that subtracted three of that away from that I guess in this case that was all so there were just two e's in this case one that did this step and one that did this step and together they gave me an e that does.
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And the net result was to get an odd and you can tell me what that has to be.
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This is like the picture of what happens if e multiplied a whatever that is we never figure it out in this way but whatever that e times that e is e times a is what's e times it.
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So that e whatever the heck it was multiplied a and produced up so e must be e a equally i tells us what e may link it in in the in verb today great.
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And therefore when the second half when it is multiplied by a c but then say it was the c the picture looking that way e times a is the identity that tells us what e has to be it has to be the
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identity and therefore on the right hand side where e where we just marked it's tucked on the identity it's turning in step by step it's turning into a
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way e there is the statement of del showing the elimination that's how you find the end where we look we can look at it as a elimination as solving in equations at the same time and
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then passing on n column solving over equations up to the n column for the n.
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Okay thank you. Thank you on the way.
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Thank you.
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Thank you.
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Thank you.
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Thank you.
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Thank you.