Birne Binegar: Okay, so.
Welcome back.
Birne Binegar: let's see i'm going to begin today.
Birne Binegar: By announcing.
Birne Binegar: That we are, we are not exactly returning but we're going to transition to face to face lectures on Monday.
Birne Binegar: University gave us two weeks to get through the.
Birne Binegar: dimension and now we're back to our regular scheduled lecture, which is a face to face in person lecture for the.
Birne Binegar: 930 section of linear algebra were to me Monday Wednesday Friday 930 1020 but here's the difference, instead of one key up on zoom your to attend class 312 in more classroom building that will begin on Monday.
Birne Binegar: Any questions on that.
Birne Binegar: Okay, now, of course, according to university policy if we have.
Birne Binegar: covered cases arise in the class at once, we met then we'll probably transition back to online only But until then deal that happened were to actually conduct our face to face lectures again.
Birne Binegar: Okay, so um.
Birne Binegar: let's see so today we're going to talk more about matrices.
Birne Binegar: On Wednesday, I introduced matrix these as a way of encapsulating data that goes into specifying a large system of linear equations.
Birne Binegar: What happened and we start off with say system.
Birne Binegar: Of equations.
Birne Binegar: In em and down so here's maybe the first equation, the one in a series of equations maybe the empty equation.
Birne Binegar: A in one X one plus.dot.us a and M and N variable is equal to am number.
Eli Lucas: The system of equations.
Birne Binegar: could be written the matrix form.
Yes.
Birne Binegar: They want one at 1am the one and and and column vector variables.
Birne Binegar: Times well equals another column vector and.
Birne Binegar: What i've been explaining started on Wednesday was how to interpret the left hand side here as a product of one matrix times the column vector.
Birne Binegar: So.
Birne Binegar: that's what we're going to begin today.
i'm.
Birne Binegar: Describing more explicitly this matrix multiplication and we'll do lots of computation So you see how this is carried out, so let me just remind you, these are rectangular arrays of numbers are called matrix tease and we've gotten location for those this is how we treat the generic.
Birne Binegar: n by n matrix matrix and rows and columns So what we do is, we take a bunch of letters.
Spencer Majma: With sub scripts and put.
Birne Binegar: them in a box.
Birne Binegar: But in a special way.
Birne Binegar: We have a total of em column sets the second number here and a total of in rows.
Birne Binegar: And so, this matrix is a rectangle arrangement of total and times in real numbers into in rows and columns.
Birne Binegar: And notice that these in this generic location for our matrix that i'm always using the first subscript the first index to indicate which rowan entry lives in the everything in the first row.
Birne Binegar: Yes, a subscript one way, and the second index is used to indicate which column that lives in those is sort of like when you're grabbing things first component is horizontal component second component is worth the same idea but convention that we keep throughout.
Birne Binegar: Alright, and then what we began to discuss on Wednesday was how to multiply these objects together.
Birne Binegar: So, in order for a matrix multiplication to be defined the crucial thing is that the first factor, a has to have the same number of columns as the second factor, he has wrote.
Birne Binegar: But so long as that is true what's going to happen if you can define the product we're in the product of a n by n matrix a in an invite P matrix be so and has in rows and columns second factor as m rose and pecans.
Birne Binegar: In this situation, the product is.
Paxton Krafft: Sorry.
Birne Binegar: Is there a question.
Birne Binegar: Okay, so in this product.
Birne Binegar: The way we compute it is we interpret a row, is an ordered list of numbers so it's a factor.
Birne Binegar: And we can also interpret a column as an ordered list of numbers read from top to bottom, and so we can think of both.
Birne Binegar: The rows of a in the columns of be as being vectors and, in fact, since they have the same number of entries, there are going to be vectors of the same dimension, so we can take the Doc product and that's exactly what we do.
Birne Binegar: Right and then that dot product okay.
Birne Binegar: A certain dot product, one for each row of a in one for each column of be paired them up this way compute those numbers.
Birne Binegar: On the other hand, if the second matrix be doesn't have the same number of rows as the, the number of columns of the first factor, then that would we try the same formula that would mean trying to take a.
Birne Binegar: row vector one dimension and taking it dot product with a vector will have a different dimension and that's undefined until for this matrix product to actually work.
Birne Binegar: it's essential that the number of columns in the first factor equals the number of rows of the second factor that just show these Doc products already fine, but once you define their computer.
Birne Binegar: Now, I think the point out also here is that coming from this world we get another world.
Birne Binegar: And this line here i'm what i'm describing are the dimensions of the matrix and rows and columns that's the dimensions of the first factor time to matrix with.
Birne Binegar: em rows and columns what's that going to produce is a matrix with in rows and P called why because we look at these entries that we can compute.
Birne Binegar: What are our choices for this index on well it's gonna be a row with the first matrix a and he has exactly in rows so this index I we have exactly and choices.
Birne Binegar: and similarly for this index J well that Jay refers to a call of the matrix be be has exactly pecans so we'll have P choices for Jay, and so what I get is a bunch of numbers on a the subscript I J or which we have.
Birne Binegar: invoices, for I P choices from Jay so we're getting a range those into an end by P matrix and that if that's exactly what the product of the matrix ad is going to be.
Birne Binegar: alright.
So let's do some explicit combinations now.
Birne Binegar: The first example is kind of a non example i've gotten a one by three matrix written adjacent to a matrix with three rows and two calls.
Birne Binegar: But this matrix as a as a thought of the product is undefined let me show you why here we have.
Birne Binegar: A matrix with three rows in one call so with three by one matrix it's trying to multiply a matrix with three rows and columns so a three by two matrix.
Birne Binegar: But notice this mismatch on these inner dimensions, you need to have this match, and so that the dark products that we find the the matrix product are defined.
Birne Binegar: Joe because we have this mismatch that's why we have an undefined matrix product it's not computer and it makes no sense mathematically to consider.
Birne Binegar: But.
Birne Binegar: That does not mean that matrix multiplication are always undefined We just need to get this a proper match between the number of rows of the first factor, and the number of columns of The second factor.
Birne Binegar: Okay, so let me do another example where we are multiply a vector by a matrix.
Birne Binegar: Alright, so now, here, the point is is that if we think about vector is simply an ordered list of numbers, then there are two ways of interpreting this.
Birne Binegar: What a list of numbers so matrix I can either take the fb components of this factor.
Birne Binegar: keep them in order but write them, you know vertical fashion, so now, I read from top to bottom, to identify the order of these numbers.
Birne Binegar: Or, I can arrange these numbers, you know that oh sorry I shouldn't have put commas here, just as a horizontal list, and that would be.
Birne Binegar: A one by n matrix So in fact there's two different ways of taking an order list of numbers and converting it into a matrix for the most part we're going to.
Birne Binegar: stick to the first Convention, we have an ordered list of numbers we're going to write it as a column vector and it's an n by one matrix.
Birne Binegar: Alright, so let's anyway, we can have two different ways of computing vectors into maker sees.
Birne Binegar: let's see what happens when we multiply.
Birne Binegar: Each of these types by another matrix so here, I have a order list of numbers one minus one and i've written it vertically it's a column vector i'm going to try to multiply it from the lab by a two by two matrix two columns two batters.
Birne Binegar: Alright, so let's first look and see if this thing is defined well, we have a two by two matrix multiplying a matrix with two rows in the single call.
Birne Binegar: So we have the match, we need on the on the inner dimensions.
Birne Binegar: And what's going to get kicked out by the product is going to be a two by one matrix so you need this match on the inner dimensions for it to be defined, and then the result is going to be well sort of collapsing, of the outer dimensions.
Birne Binegar: Right and now.
Birne Binegar: So we got the right condition for the matrix product you'd be fine now we can compute it alright so what's going on here, let me go to the pad of paper.
Birne Binegar: Here we're let's take a equal to.
Birne Binegar: One two minus one, two and this isn't the is going to be a column vector one minus one still matrix writing this gun goes vertically.
Birne Binegar: Now the product, a B.
Birne Binegar: should be a matrix with well it's going to be two by one matrix.
Birne Binegar: That means it's going to have.
Birne Binegar: Essentially, two components in the first row and first column it's going to be the first row of the first actor.
Birne Binegar: got product with the one and only column of the second factor be alright, so what is that i'm competing in here.
Birne Binegar: here's the first row one, two, I interpreted as an ordered list of numbers and i'm going to take it Doc product with this ordered list of numbers.
Birne Binegar: Written vertically here, but here i'm just treating it as an ordered list of numbers, and that gives you one one entry and add the entry in the second row in the one and only column at the product is going to be the.
Birne Binegar: Road to have the first factor.
Birne Binegar: doctrine it.
Birne Binegar: First.
Birne Binegar: And only comb The second factor.
Birne Binegar: Right and so that's what i've got written on the right hand side here here's the second row minus one two now i'm thinking of an ordered list of numbers, for which I can take it Doc product with another order list of numbers of pain from this column that.
Birne Binegar: Alright, then we compute those products right one punch one is no here, plus two times minus one.
Birne Binegar: And then the second row is going to be this component this component second component time the second component.
Birne Binegar: And only blessedly do the arithmetic here we get minus one minus three of the result, and indeed this product of a two by two matrix in a two by one matrix produces a matrix with two rows in a single call.
Birne Binegar: Alright, so that's when I interpret the orders list one minus one as a column matrix.
Birne Binegar: But I can also take that same list of numbers one minus one and interpreted as a row matrix let's compute the product there oh one thing I should point out that.
Birne Binegar: So here i'm taking one and minus one thought of it simply as a ordered list of numbers, you can see that.
Birne Binegar: And it can either go.
Birne Binegar: To this vector this column vector or this robe after.
Birne Binegar: we've looked at the product of this vector with that two by two matrix now i'm going to show you how to multiply this guy times the same matrix, but one thing that's going to happen here is.
Birne Binegar: The actually the order of the factors is actually already prescribed by these dimensions.
Birne Binegar: We go back to the.
Birne Binegar: original case.
Birne Binegar: We do have.
Birne Binegar: let's see one two.
Vivian Taylor: If the paper down a little bit, please.
Vivian Taylor: We do have.
Birne Binegar: This product is defined one minus one so when I write this vector originally one minus one it's an ordered list and I write it as a as a column vector this matrix is defined and it kicks in what it is that kicked out.
Birne Binegar: Number minus one minus three Is that correct.
Birne Binegar: yeah so that would be fine, but if I tried to put one.
Birne Binegar: minus one on the opposite side of one two minus one, two, well, here we have a two by one matrix trying to multiply a two by two matrix and we have this mismatch.
Birne Binegar: Right.
Birne Binegar: Because the internal dimensions on not the same i'm not gonna be able to take Doc products or.
Birne Binegar: rows of the first factor with columns of The second factor until this is undefined.
Birne Binegar: Okay, so.
Birne Binegar: Somehow some products are going to be defined, some products are undefined you have to look at the dimensions of the matrix to decide how to multiply things together.
Birne Binegar: Alright, so Now let me turn to the second way I can break this guy not only as a vertical vector but also as a row vector so i'm going to write it as a row vector that's going to force me to actually multiply this two by two matrix from the.
Birne Binegar: left by this row vector because, otherwise, but the product of the row vector times the matrix is not going to be fine.
Birne Binegar: I need the same number of columns of the first factor I need that the match number of rows of the second factor in order for the product to be defined alright so i've done this correctly and now i'm going to compute do I take the dot product first row.
Birne Binegar: Of the second factor with the sorry the first column The second factor with the first and only row of the first factor that's this entry here.
Birne Binegar: And then the second entry it's going to correspond to taking the second column of the second factor and taking its dot product with the first row vector of the first actor that's this entry you don't want to compute.
Birne Binegar: We get the know the vector to zero as it's appropriate, because what we're now doing is taking it to buy one sorry a matrix one row in two columns multiplying two by two matrix.
Birne Binegar: And what is kicking out is one by two matrix matrix you have a single row and two columns as know exactly what popped out of this come computation.
Birne Binegar: So when we compare these two results, we had a we can take this.
Birne Binegar: Order list of numbers one minus one and write it as like a column matrix and then I can take it product with this square matrix this two by two matrix and I get the result liners one three.
Birne Binegar: Or, I can take that same order list one minus one right it as a robot or roaming drinks.
Birne Binegar: And I can compute its product with the same matrix because of dimensions have changed, I have to actually multiply from the left by this matrix now for that this product, and the second line to even be defined, but what it is defined by this arrangement and upon computation I get.
Birne Binegar: A real vector with entries two and zero, and so what you see is that there's not only two different ways of converting order list of numbers into a matrix but, in fact.
Birne Binegar: When we start doing matrix multiplication it's going to make a difference, as to whether we interpret a vector as a column matrix or Roman.
Birne Binegar: know what.
Birne Binegar: What in fact happens is.
Birne Binegar: Treating vectors ordered list of numbers that's calling vectors or rogue actors that's a matter of convincing well for the most part it's a matter of Convention.
Birne Binegar: We will do in this course, this will constantly think of that what our list of numbers, as most directly corresponding to these calm actors.
Birne Binegar: were just adapt that Convention, we could have also insisted on only using row vectors to represent ordered list of numbers, but then.
Birne Binegar: we'd have to do everything over again, because now, the ordering has been changed, so what we do, primarily in this course in this regard ordered list of numbers.
Birne Binegar: As column vectors and we, for the most part stick to that Convention and we only interpret order list of numbers is rolling as rope actors only once special circumstances justify that okay.
Birne Binegar: So in any way, we have a choice of whether we interpret an ordered list of numbers, a row vector a column vector will tend to use that convention vectors go to college lectures and then I just want you to be aware that it does make a difference.
Birne Binegar: All right, let me show you.
Birne Binegar: Another.
Birne Binegar: property of matrix multiplication here i'm going to start with two two by two matrix see both these major season two rows and columns and their product it's going to be defined, why.
Birne Binegar: If I have a two by two matrix.
Birne Binegar: And it multiplies another two by two matrix we have the match, we need.
Birne Binegar: That means it's defined.
Birne Binegar: And just define what's going to get kicked out so by two by two, so what you do is you check the map and what you do is you check make sure that the inner dimensions match, and then the outer you mentioned.
Birne Binegar: they're going to tell you what the dimensions of the resulting product is going to be, it will they be defined is going to have these dimensions.
Birne Binegar: Right so let's do the explicit computation for this to these cases, here we have to one.
Birne Binegar: minus one one.
Birne Binegar: minute you're going to multiply one minus one.
Birne Binegar: minus one sheet alright, so the rule is, let me write it out enlarged form, so this is the.
Birne Binegar: The entry in the first row and first column, so this is the.
Birne Binegar: Only we just call it a.
Birne Binegar: b one B then we'll have another entry a B one to have a third entry.
Birne Binegar: A, B now living in the second row first call in this last entry is going to be a be lives in the second row second call alright, so this is the I think we're going to compute in our rule is and set these components.
Birne Binegar: we're about to compute are computed by taking the road, corresponding to the first index, the first factor, and we take its dot product with the row oh backlink the column, corresponding to the second factor so i'm starting to shoot in column J second factor all right let's.
Colin: move on your maker little bit, please.
Colin: The better Thank you yeah.
Colin: let's look at.
Birne Binegar: For example, a B one one now what's the first row of a it's two one.
Birne Binegar: what's the.
Birne Binegar: what's the first call up the second factor well that's.
Birne Binegar: i'm going to write it as an order lunch i'm no longer right it vertically minus one minus one now, I have two two dimensional vectors but which I can kick Doc product one time two times, one is two.
Birne Binegar: plus one minus one which is minus one before we're going to get one to what I can do just replace that me.
Birne Binegar: In the one one slot I can replace it with the value of this component, there will be one.
Birne Binegar: it's all you can do that what you have to do is separately take each row of the first factor and separately take its job product with each column The second factor and.
Birne Binegar: That will fill out this matrix i'll let me compute one more let's do a B now look in the first row of the second column result we're still using the first road a but now we're using the second call of the factory T.
Birne Binegar: minus one, two and that got product it's going to be two times minus one negative two one punch to is to those two numbers something zero until that's why I got zero in this slide.
Birne Binegar: All right, then, then, to get the minus two what you would do is you take.
Birne Binegar: Second it's in the two one slot of the product and it's going that's going to be a pain, by taking the dot product of the second row of a with the first column.
Birne Binegar: B and not going to British minus two.
Birne Binegar: right then.
Birne Binegar: Both of these makes received or two by two matrix ease and the opposite product be a is also defined because I had thing number columns of the first factor is I have rows of the second factor.
Birne Binegar: But now, when I computed, for example, one minus one dotted with two minus one is going to produce the number three.
Birne Binegar: and, similarly, we use this formula to calculate the other entries of the matrix product and Lo and behold the product of a times B is different, that result is different than the product that be tied to a so what products are defined, but they yield different matrices.
Birne Binegar: So that's an important property it's actually part of what makes matrix multiplication useful because it's more subtle than simply.
Birne Binegar: The multiplication of numbers now with multiplication of numbers two times a is equal to 16 but a conch to is also equal to 16.
Birne Binegar: When you multiply numbers it doesn't make any difference that the order of the factors but major sees the order of factors is actually makes a big difference just matrix is nothing like the second matrix.
Birne Binegar: Okay, in fact, it's even worse, not only these different sometimes one product is defined in another product is not defined the products in the opposite order is not defined.
Birne Binegar: So, for example, if you have them invite him matrix in rows and columns for the matrix day and you have an mvp matrix wealth and this product is going to be defined.
Birne Binegar: You have the match of the inner dimensions, so that means it's defined, and then the outer dimensions are going to give you the result of the actual computation it's going to be an invite P matrix when we compute a times B.
Birne Binegar: However, if I tried to compute the the opposite product and like put the factors in the opposite order now i'd be multiplying.
Birne Binegar: Invite P matrix lunch n by n entries and that's generally going to be undefined because the only one times this product it's going to be defined, is when the inner dimensions match.
Birne Binegar: And that's not going to be the what happens usually, when I reverse the order is only going to happen in.
Birne Binegar: The number of columns of be was actually equal to the number of rows of a.
Birne Binegar: Okay, so what this means is that, when we manipulate major seizing the moment multiply them, we have to pay attention to the word.
Birne Binegar: and
Birne Binegar: This is going to come up subtly and as soon as we start using nature's medication because suppose we had two matrices a is equal, but two messages on both sides, and I want to multiply this equation by another matrix so I multiply it by be well.
Birne Binegar: If you do this say that because.
Birne Binegar: A is equal to see it necessarily follows that when I multiply both sides by me, I should get the same result Well, this is not true.
Birne Binegar: The reason why it's not true, in general, is because I multiply a bunch of left by be but i'm applying the other side of the equation from the right by be and these.
Birne Binegar: even have a is equal to see these products will not be the same because one on both on one side and multiplying from the lab, on the other side i'm multiplying on the right.
Birne Binegar: So the warning i'm going to give you is that, when you start manipulating equations you have to make sure you're doing exactly the same thing to both sides of the equation, otherwise you're going to introduce an error.
Birne Binegar: In error your calculation.
Birne Binegar: If this equation is true, and you want to maintain it after performing an operation on both sides of the equation, you have to do exactly the same thing to both sides of the equations.
Birne Binegar: All right, and multiplying from the Left it's not the same thing is multiplying multiplying from the Left it's not the same thing multiplying from the right.
Birne Binegar: Okay, so let me show you.
Birne Binegar: outfitter undefined example be correct Well now, the point is.
Birne Binegar: If a product is undefined all the correct statement is done it's undefined it's that life taking three and five divided by zero it doesn't equal anything legitimate mathematically and so you just say that 3D zero divided by.
Birne Binegar: Three divided by zero is undefined that's in the same sense that i'm.
Birne Binegar: A product of two matrix ease, in which the.
Birne Binegar: On dimension of the first matrix is not equal, the road dimension of the second matrix that it's going to be undefined just we just don't have a definition for that product.
Birne Binegar: But.
Birne Binegar: When the matrix multiplication is defined it turns out to be really useful so we're not going to worry too much about the undefined cases the cases were matrix multiplication is useful, is exactly the cases when the multiplication is defined.
Birne Binegar: Alright, so here, I want to point out in this slide that I take two real numbers and I multiply them.
Birne Binegar: together and I get zero well, I know, then that either X has to equal to zero, or the second factor, why has to equal zero otherwise it's not possible to take to non zero numbers multiply them together and get zero.
Birne Binegar: But look at these matrices they're both two by two matrix ease and their product is defined but they're not zero matrix yet their product is zero matrix in here.
Birne Binegar: What I mean by the zero matrix you know, similar to the one i'm at by the zero vector it's the matrix for which each entry is equal to zero.
Birne Binegar: Until here, we can have two non zero matrices multiplying, it will play together producing a well defined product but it happens, the equal to zero matrix which is not unlike what would happen if we regarding these matrix aegis numbers.
Birne Binegar: Alright, so.
Vivian Taylor: So if we get a zero matrix is that, like an empty set, or do we just like input it as all those zeros.
Birne Binegar: know it is a.
Birne Binegar: matrix it's an ordered list of.
Birne Binegar: itself arrangement of numbers into a rectangle arrangement, so it is, it is a matrix and it's not it's not empty it's something it is.
Birne Binegar: A gentleman tube this 0000 matrix is a legitimate two by two matrix i'm just saying, if I take this matrix in this matrix I get this matrix.
Birne Binegar: what's bizarre is that if these were numbers, the only way I could get the numbers zero is by taking one of the factors equal to zero.
Birne Binegar: But here i'm saying that I can get the zero matrix by multiplying together to nonzero matrices that's strange, or at least it makes matrix multiplication different than multiplication of numbers.
Birne Binegar: And the reason for pointing this out is that no occasionally.
Birne Binegar: we're going to learn that well right matrix multiplication exactly like.
Birne Binegar: When we multiply numbers together or write something adjacent to another, but what i'm trying to point out is that these products behave different than the products of numbers.
Birne Binegar: Okay, so that that's the point i'm making Is it because her new location is similar I want to stress how how things are different when we have multiple issues together as compared to when we multiply numbers together.
Birne Binegar: All right.
Now.
Birne Binegar: There are some things that are similar to multiply numbers, if I have a number eight and multiply it by the number one I always get the number eight back, we call one, this is the multiplicative.
Birne Binegar: identity.
Birne Binegar: Well, when you take them an abstract algebra course if you have something that you multiply another thing in it, but it always reproduces what you started off with we call that an identity operation.
Birne Binegar: and multiplication by one is an identity operation on the set of real numbers well there's an identity operation also on matrix please.
Birne Binegar: don't let me show you an example, but take this matrix it's going to be a three by three matrix where the only non zero entries how long this diagonal from the upper left to lower right.
Birne Binegar: Now i'm going to take an arbitrary three by three matrix so i'm not even going to specify this particular numbers and these nights lot of there's three by three matrix.
Birne Binegar: running these guys adjacent to another you Steve we have a three by three multiplying and three by three.
Birne Binegar: That it's going to lead to a well defined matrix product, because we have the same number of columns of the first factors we have rows of the second factor.
Birne Binegar: But when we computed well let's compute this first one one entry, it would be the dot product of 100 with the call a one one a 21831 until will have one time at one zero times a one two plus zero times 831 well, that will be.
Birne Binegar: A one one plus zero Plus zero, and so it just just going to replicate that first slide.
Birne Binegar: By the way, let me, let me point out in this example, how to do these matrix multiplication for the most part.
Birne Binegar: yeah we have 10001000 Horn opinion and we're going to multiply a one one day 123 in q1 and q2 q3 a real one, a three to the.
Birne Binegar: Let me show you how you do this real fast right so just with your fingertips move across.
Birne Binegar: move across the first row and then all of a second finger on move down the second, the first call so i'll have one one times a one.
Birne Binegar: Plus 00 times anything is going to equal zero Plus zero, so this is going to be one plus zero Plus zero I can just see it visually one plus zero Plus zero the second in the second slot in this.
Birne Binegar: First row second call i'll take the god product this first row with the second column that just do it in place on him one time a one two.
Birne Binegar: Plus zero Plus zero so look where my fingers are and then multiply them again now, the this product will be 812.
Birne Binegar: second term will be zero.
Birne Binegar: And the third term is going to be zero and so yeah i've done the first.
Birne Binegar: The first two entries of the first row of this product, but you should be able to do this fairly quickly you just read across and.
Birne Binegar: compute these products and then add them to suck.
Birne Binegar: All right, and then Similarly, I can also, since these are both three by three matrix ease I can do this multiplication you know it's well defined in either order and, in fact, in this instance.
Birne Binegar: The order doesn't seem to make any difference.
Birne Binegar: This matrix times this generic matrix produces the generic matrix again, even when I even when I do the the multiplication in the opposite order.
Birne Binegar: So for this particular matrix little tall I anytime I multiply a three by three matrix diet, whether I multiply from the right from the Left, I just get back to the matrix I started off with.
Birne Binegar: Until effectively this matrix behaves like the number one with respect to matrix multiplication that's why we call it the the identity matrix.
Birne Binegar: And a generalized is to larger square matrices larger make sure she's wearing have an equal number of rows and columns.
Birne Binegar: You created by an identity matrix by simply putting one too long, this diagonal from the upper left to the lower right and zeros everywhere else, and then that matrix is going to have the property that whether I multiply another matrix by it it's always going to kick back.
Birne Binegar: that's.
Birne Binegar: The matrix so eight times is going to equal a whenever this product is defining so, no matter how many rows and columns.
Birne Binegar: A heads, as long as this product is defined it's going to kick back a and, similarly, when I multiply from the left by this identity matrix so long as the product is defined it's just going to kick back the second factor.
Birne Binegar: Okay.
Birne Binegar: So.
Birne Binegar: that's how we multiply two matrices together so we'll have lots of opportunities.
Birne Binegar: You practice them in this, and, in fact, let me remind you that homework assignment one on website is due at midnight tonight.
Birne Binegar: And what i'll do after class is i'm going to assign me web assign problems that too and it's actually it will be do a week from today, and it will do with this matrix multiplication and some additional matrix operations and about the cover.
Birne Binegar: Okay, so.
Birne Binegar: Up to this point in today's lecture i've been talking about multiplying one matrix times another matrix now i'm going to tell you about some other things we can do the matrices we can, for example, we can add them.
Birne Binegar: And this edition of matrices it's going to look very much like the way we add vectors.
Birne Binegar: You i'll show you in the manner which that's true when you add vectors you add like components to like components.
Birne Binegar: The first component is added to the first component to get the first component of the sun, well, we can add matrices by a similar rule will instead of component.
Birne Binegar: Where i'm just thinking of an ordered list or refer to the numbers that appear in the matrix all refer to them as entries.
Birne Binegar: And so here, what happens if you can some two matrices so long as her matrix ease of the same size same number row seat number of columns.
Birne Binegar: Then I can add them together by simply be I jade component, the element and the iPhone jade calm and the sun is going to equal the some of the corresponding component in the individual major cities.
Birne Binegar: So here's an explicit example to know let's say.
Birne Binegar: 211 minus one plus 1002 that's going to be exactly the matrix two plus one is three one plus zero is one one plus 01 and minus one plus two is one.
Birne Binegar: Okay, so you just got to do that sort of like adding vectors except now you have to track all the entries in the matrix and put their their thumbs and the same slot.
Birne Binegar: Alright, so that matrix edition.
Birne Binegar: and
Birne Binegar: For vectors we'd have a notion of scale or multiplication we also have scale and multiplication of matrices it's essentially a simple rule.
Birne Binegar: The skill or multiple of an n by n matrix by a number Lambda is going to be the N by n matrix which the entries of a have effectively been all multiplied by the same number Lambda so.
Birne Binegar: Personally, I want to know it two times one zero minus one three is it's going to be exactly the matrix to two times zero is 02 times minus one negative to two times three is six.
Birne Binegar: So we have matrix multiplication and and sorry scale of multiplication in addition to matrix multiplication.
Birne Binegar: And later on when we start to consider more general vectors faces will be able to view nature sees as a kind of vector the reason why we can consider major season vectors is because.
Birne Binegar: Now we're thinking of a more general definition of what a vector is now.
Birne Binegar: Instead of thinking of a vector as an ordered list of numbers that's going to generalize to set for which we have the gentleman operations of.
Birne Binegar: matrix vector addition in scale multiplication here's this is going to be vector addition when I interpret matrix these.
Birne Binegar: abstract vector, and this is going to be scalable application when I interpret a matrix as an abstract vector not so we're going to list of numbers but it's something that behaves like organisms numbers okay so.
Birne Binegar: Right now, this is going to be confusing is the Why would want to think of a matrix as a vector that's only going to become.
Birne Binegar: Later in the course when we even think of sets of functions as vectors we think of oscillations is vectors we think of states of the hydrogen atom and the vectors that's when we think of more abstract object if you having like vectors.
Birne Binegar: But here at this stage of the game, I just want to point out that you can add matrices together and you could scale multiply them.
Birne Binegar: To login operations that we can perform on major seeds that are distinct from the matrix multiplication remember have been talking about previously in this lecture.
Birne Binegar: So.
Birne Binegar: So, here are some simple examples of matrix edition.
Birne Binegar: Just like to add, before you add an entry by entry to get the result position, and when you do scale the multiplication you multiply each entry of the matrix by the same number.
Birne Binegar: Alright there's Another thing we can do the matrix.
Birne Binegar: If you have an n by n matrix you can convert it to.
Birne Binegar: n by n matrix what you do is you take rose.
Birne Binegar: And you can.
Birne Binegar: send your top of the page sorry.
Birne Binegar: yeah better what we'll do is we'll convert rose in the carbs well, let me give you an example to 11011 alright, so this matrix has two rows and three calls where i'm gonna do is i'm going to convert it i'm going to take this row.
Birne Binegar: i'm gonna write it the numbers from the same order, but instead of writing and horizontally and vertically and similarly that second row if I convert it to a call i'll get this and so what's happened I converted a two by three matrix two or three by two matrix.
Birne Binegar: This operation that we can perform on any matrix is called the transpose operation.
Birne Binegar: So, here it uranus defined here's another way of defining it.
Birne Binegar: The transpose of a matrix hey yes, the matrix you get by.
Birne Binegar: Effectively, switching rows and columns but here i've written in terms of individual entries in here what you see.
Birne Binegar: Whenever I write down a matrix the first index always corresponds to a row index, and the second index corresponds to a column index so for this transpose matrix.
Birne Binegar: The entry in the IGF slot is to be the entry so at the end, fearing the I wrote and Jay called the transpose is going to be the entry in the Eighth row, and I call them at the original matrix so what happened here this entry in this.
Birne Binegar: here's the transpose this entry in the first sorry second row of the first column of the transpose is equal to the entry in the.
Birne Binegar: Second column of the first room of the original matrix so rows and columns that interpretation gets reversed when we take the transpose.
Birne Binegar: So here's an explicit example here's a matrix that's now a three by two matrix and when I take it to transpose it gets converted into a two by three matrix.
Birne Binegar: So.
Birne Binegar: Now here's one thing that transpose operating is good for if you start with a.
Birne Binegar: matrix with in rows and columns well then it's square it's not necessarily defined right because.
Birne Binegar: suffocated here.
Birne Binegar: If I have an n by n matrix and I try to multiply itself, which is still in in on him matrix I get a mismatch.
Birne Binegar: So this product is not going to be defined, so if this is a, this is a n squared is undefined.
Birne Binegar: i'm computer.
Birne Binegar: So you can always square matrices, however, that turned out that there still an analog of taking the square of a matrix it's just that I have to multiply it by its transpose and, in this case i'm.
Birne Binegar: A i'm a transpose this would be something like in my m notes major now i'm going to take the transpose I get a matrix where they rowan column dimensions have been reversed Now this is defined, because we have to match me and what's going to get kicked out it's going to be in by n matrix.
Birne Binegar: But we can also do it from the opposite side as well, we can take the transpose then multiply the original matrix buy it now, in this case we're taking them n matrix that's the dimensions of the transpose and we're multiplying and N by n matrix, this is still defined.
Birne Binegar: But for this product we're going to get this result and and by me trees so.
Birne Binegar: We can, in general, take the square for matrix but with the transpose operation we have two analogs of the taking the square we either multiply from them right by the transpose.
Birne Binegar: Or we multiply by the left of the trains posed unfortunately we get different results, but we have two different products that we can associate with.
Birne Binegar: Playing essentially the analog of the square of the number you're I can always multiply Well, I can always multiply a by itself, I can always multiply a by its trench coats.
Birne Binegar: All right, and then enter in a particular case is when you start off with the column vector and then that means that the product and have a column vector Now this is going to be a matrix so I can.
Birne Binegar: The gop product, the column vector times at home that you're generally going to be undefined right by having.
Birne Binegar: A matrix with inroads in a single call and I tried to multiply it by itself, well, we have a mismatch.
Birne Binegar: So let's in general undefined I can't multiply column vectors times column actors, however, if I take the transpose.
Birne Binegar: of an n by one matrix that becomes a one by n matrix and that so after and take the transpose I reverse the dimensions and now I do get a product that's defined.
Birne Binegar: And in fact what that is it's the usual dot product of this column vector when you when you think of it as an ordered list of numbers, you see.
Birne Binegar: When I take this transpose of this column vector and I multiply the from the left by the transpose this is the matrix product and that's going to be exactly equal to the Doc product of the thought of as an as an ordered list of numbers, with itself.
Birne Binegar: tolling fact or non product, the vectors in matrix language actually corresponds to taking interpreting the vector as a column vector and then calculating it's dot product itself like first converting.
Birne Binegar: or forming the be left hand factor is a transpose of the original column vector the.
Birne Binegar: OPS i'm sorry and i've run three minutes over so i'll stop here you have questions i'll hang around a little bit, but other than that classes this this but remember on Monday we're going to meet in the north classroom building at 930 Okay, so I will see you there.
Thomas Kidd: Is.
Thomas Kidd: This is the a vector.
Thomas Kidd: or sorry a matrix equal the a transpose matrix or no.
Birne Binegar: Oh, how the point of the transpose operation is it changes the matrix.
Birne Binegar: matrix within rows and columns and it sort of.
Birne Binegar: converts it to another matrix of with different dimensions and sort of the opposite dimensions but it's easy to do the operation that's right a B, C D.
Birne Binegar: Well, when I take it transpose rule is just that you pick a row, and you rewrite it as a column take the second row, and you rewrite it as a column, so this matrix has changed.
Birne Binegar: And so it's just an operation that we can performance now.
Birne Binegar: What it's good for that that's coming later i'm just want to make you aware of things we can do to make your season.
Birne Binegar: Oh there's another question that.
Birne Binegar: says will lectures.
Birne Binegar: generally be recorded when we meet in person ah.
Birne Binegar: Yes, and no, I want to get away from recorded lectures, because I don't want people thinking, they can avoid lecture it's very important to come the lecture and participate in the class so i'm gonna.
Birne Binegar: I think next week I probably am going to record because.
Birne Binegar: anytime you feel ill I don't want you to come into class, in fact, you should not be coming to class if you feel the onsite or something that just going to shut us down for two weeks, but.
Birne Binegar: So long this word.
Birne Binegar: The coven situation is under control, the intent is to actually go back to normalcy it's in some form until I want to get away from recorded lectures, because what happened during the pandemic is.
Birne Binegar: First of all, people stopped attending the lectures, so you occur and then.
Birne Binegar: Once they realize they can get away with that they postponed even looking at the lectures until right before the exam.
Birne Binegar: And that's just a disaster for the success rate in this course so i'm going to try to force you guys to come to the lecture so that we can actually get some learning done in class that will save you time.
Birne Binegar: In prepping for exam and doing homework problems so next week, yes, I will record the lectures and make them available but i'm going to be more restrictive as soon as the closer we return to normalcy okay.