Birne Binegar: Okay welcome back.
Birne Binegar: Today, the first thing I want to do is give a quick demonstration of how to.
Birne Binegar: carry out your web assign problems me change the screen share here.
Birne Binegar: Okay, so what you should be seeing on screen is effectively.
Birne Binegar: How well the sign should appear to you once you've logged in.
Birne Binegar: To the appropriate section, so now, by the way, to access web assign I think you get the first two weeks free but eventually, you have to purchase subscription.
Birne Binegar: And the class key for this section of linear algebra is OK state 29201911 that's the information you feed web assign once you've subscribed, and that will give you access to the assignment assignment number one.
Joseph Behm: is going to be do.
Birne Binegar: This Friday fairly quick assignment issue assume see but.
Birne Binegar: Anyway, one to.
Birne Binegar: gain access to web assuming this is what your screen should look like and to work on any particular problem said click on the assignment.
Birne Binegar: goes and.
Birne Binegar: You typically will answer things by filling out these web forms.
Birne Binegar: So, for example, let's first problem.
Birne Binegar: Your to identify the figure that this displays a doctor who's tail is at the point three one and whose direction is heading in the direction five corazon Units one vertical unit so let's see.
Birne Binegar: This looks like it, starting at the point three units over.
Birne Binegar: One unit up and.
Birne Binegar: Unfortunately it's going five minutes in the vertical direction, rather than the horizontal direction so it's actually this last picture of its best represents this situation, a vector that starts at three one and heads in the direction by zero.
Birne Binegar: answer the person problem for you, although.
Birne Binegar: The individual problem sets that you students work out are going to be slightly different because, whether sign randomized all its problems, and so.
Birne Binegar: problem one one a that i'm sure display now may not be exactly the same problem you see this once you've logged into web assign and start working out the problems, but it'll be the same type of problem The other thing I wanted to show you is.
Birne Binegar: How to enter vectors so let's user most.
Birne Binegar: picture problems, but here, here we have the problem for we're told the end two vectors.
Birne Binegar: Now, whether sign is writing the vectors vertically, in most cases, so it's still been ordered list first element in the list for a is three The second element is zero and we can still use the rules for combining.
Birne Binegar: vectors why addition use by adding like components what's going to happen in this initial problem is.
Birne Binegar: To get the first component of the sound you add the first component of the individual summon so three plus three is going to equal.
Birne Binegar: Six and then to get the second component.
Birne Binegar: Zero plus four is going to equal for until that's how you will enter the answer the problem for.
Birne Binegar: me here for things where you just select graphical I think that that's good enough so.
Birne Binegar: The thing I want to clue you in about is that when you enter vector results you can enter them component by component there'll be a separate text entry box for each component of your answer if your answer is a factor.
Birne Binegar: Okay, so I want to show you that much and.
Birne Binegar: You should be able to finish homework set one oh 1020 minutes, or so it should be fairly easy to do and an angry it's not due until midnight on Friday, so if you do run into difficulties, we can address those in class on Friday.
Birne Binegar: Night let's now return.
Birne Binegar: To our regularly scheduled lecture.
Birne Binegar: Today, what i'm going to talk about is.
Birne Binegar: A fourth bit of apparatus that we're going to use all the time in linear algebra that is the notion of matrices but i'm going to introduce matrices slowly.
Birne Binegar: i'll show you how they come up and then eventually i'll show you what we use them for and how to multiply them together and we have a whole.
Birne Binegar: set of operations at will call matrix algebra that will be applied to these objects matrices.
Birne Binegar: But now mobile will begin today talking about what we.
Birne Binegar: discussed at the end of the lecture on Friday these geometric constructions of hyper planes i'll relate those the linear systems and then finally i'll bring matrices into the game.
Birne Binegar: Okay, so hyper planes are the key geometric construct yellow use over and over again in linear algebra their sets of vectors.
Birne Binegar: There such a factors that are formed in a particular way what you do is to create a hyper planes you start with a point in a so he's hyper planes are going to live in some vector space so ah, you can think of is be some subset.
Birne Binegar: Of the set of all possible this in numbers and the way we're created us will take a vector P zero.
Birne Binegar: Inside aren't in, and that will cause starting point.
Birne Binegar: And from that we allow ourselves to stray along in this instance K different directions, the directions, are going to be vectors indicated by.
Birne Binegar: letter D, with the subscript so notice i'm using boldface fun when I want to talk about vectors and i'm using undecorated except for their sub scripts and decorate letters to indicate real numbers.
Birne Binegar: So the way that I hyper planers forms you start off a BA n dimensional vector and then you add to it multiple have some other vectors.
Birne Binegar: This instance K different back vectors labeled by D one two DK and what that's going to happen if you very these scale or multiples two months into tk is you're going to get an infinite set of factors.
Birne Binegar: That geometrically are formed by starting in a given point and then straying away from that point in a bunch of different directions.
Birne Binegar: Alright, so that's the hyper planes, what I want to show you is how they arise in linear algebra and the first instance that will see them is when we try to solve linear equations.
Birne Binegar: So let's start off with a simple linear equation, what makes it linear is the fact that the variables X and y occur in the equation either multiply by constant.
Birne Binegar: or in separate terms, so the X variable is separated from the library bowl in this equation that's key to making it linear in X and y.
Birne Binegar: And also there's no X squared term there's no square root of X there's no either the X it's just X by itself or.
Birne Binegar: More variable multiplied by a number that's what's allowed in the equation, then you can have a term doesn't involve the variables at all well there's a linear equation, by the way, it's called linear because when you graph the solution set, let me display it here you're going to get.
Birne Binegar: A function, whose graph your y is equal to one minus X I guess it why intercept is one half and his slope is minus one half so look something like.
Birne Binegar: Something like that that line y equals one hand one minus X, so the reason why these are called linear equations is because when you grab their solution set they turn out to be aligned and, by the way.
Birne Binegar: What you'll see here, you can in this first example if the line is actually hyperplasia right here's the starting point and I can depart from that standpoint, in arbitrary scaling multiple of this direction.
Birne Binegar: This far down or I can even go in the opposite direction if I multiply this direction by him and maybe.
Birne Binegar: Okay, that what i'm going to show you more explicitly that it's a hyper planes, so I don't want to draw a picture of it, I want to express that solution set.
Birne Binegar: in this form.
Birne Binegar: So it just requires a little bit of manipulation, the first thing I do is, I take this set.
Birne Binegar: Set notation description of the solution, set and I rewrite it and so that i've had two variables X and y everything is expressed in terms of X.
Birne Binegar: And then I let X very over the entire real line so i'm doing I get all the points along this line, all the points in the solution set all I did here was take this equation for why and use it to replace the white variable and then allow extra very.
Birne Binegar: Alright, and then since i'm burying X it's not any specific number, so I can even re label it as T.
Birne Binegar: These two sets are exactly the same, because the idea whether i'm using X in the second line T in the third line, the idea is to anyway very that number over the entire real line.
Birne Binegar: Since the real line is infinite i'm, thereby creating an infinite set of vectors that are filling out that line.
Birne Binegar: Now, what I do this last line is i'm taking this expression.
Birne Binegar: here's a typical vector it's key one half.
Birne Binegar: One minus T now what i'm doing is i'm writing that is a vector 01 half so, then you see just ignore the key and I have a 011 half there and then plus.
Birne Binegar: The vector T.
Birne Binegar: minus one half T So if you add these two together to 00 the T now.
Birne Binegar: And then one half.
Birne Binegar: Plus minus one half T is going to be exactly this and then what you see is that this is the form 01 half plus T times another vector one.
Birne Binegar: minus one half and now and that's why I got written this bottom line i'm taking this vector and I just simply split it apart into a part that constant or no variables here and another term that's proportional the tea.
Birne Binegar: But this last form here.
Birne Binegar: is exactly the form of a hyper plane is just that I only have one direction D one.
Birne Binegar: So now in this formulation here, this is the p zero and this vector one minus one half is the direction the one.
Birne Binegar: Okay, so one equation, and to announce it solution set is always going to be a hyper plane that is going to be a line.
Birne Binegar: All right, what about if we have two equations in two unknowns.
Birne Binegar: Now here will use high school algebra deployment solution so first of all, the first equation for X we substitute this value for X into the second equation that will furnish.
Birne Binegar: there's 30 equation, and then we solve the third equation, for the second variable why we get y is equal to one.
Birne Binegar: Well, now we know that these two equations in play y is equal to one, but these two equations also imply that X is equal to why and that tells us since wising for the one that extension equal to do in fact.
Birne Binegar: The only possible values of X and y that will satisfy this pair of equations there's going to be X equals two in y is equal to one.
Birne Binegar: I can formulate that answer as a factor first component representative, the value of X and the second component representative value why, and this is also a hyper plan.
Birne Binegar: looks like a single point but.
Birne Binegar: It might nomenclature, the single point is also hyper plane general hyper plane.
Birne Binegar: looks like this.
Birne Binegar: plus the jade he gay when I teach for the student.
Birne Binegar: She can range over the entire life that's a general I complain, but a special case.
Birne Binegar: would be simply taking P zero and not considering any any additional directions.
Birne Binegar: So there is no the one to bk but they're not that's not necessary to be a hyper plane, this is the most general form but by taking the directions equal to zero, a single point is also an example hyper plane.
Birne Binegar: Until my shaman that solutions of linear equations are always hyper planes just maintain.
Birne Binegar: don't more generally.
Birne Binegar: we're going to consider large systems of equations meaning, we have many equations and many unknowns but they're still going to be linear.
Birne Binegar: linear meaning to the variables, as they occur in any individual equation, are going to be separated from one another in will only occur to the first power.
Birne Binegar: The only other thing that can incur in touch equations are constants that do not involved the variables at all and.
Birne Binegar: or usual notation verte equation is to put the content on the right hand side and the expressions involving the variables on the left hand side you know arranging them in this Nice.
Birne Binegar: Nice fashion listening the equations one by one.
Birne Binegar: Okay, so that's what we'll refer to, not as a system of any equations in Mr known, but all beautiful and refer to it as an n by n linear system.
Birne Binegar: Some more nomenclature.
Birne Binegar: These numbers that are multiplying the variables, those are going to be referred to as coefficient of the equations.
Birne Binegar: And in addition to that nomenclature, is this convention, where we label them these quotations the individual coefficient by certain sub scripts the first subscript I were given coefficient a DJ.
Birne Binegar: That tells you which.
Birne Binegar: Which equation urine so notice that, in the first equation, the coefficient H I J, they all begin with the subscript one second equation all the coefficients the Ai geez they're all begin with the subscript to.
Birne Binegar: Alright, so in these coefficients we kind of label, where the coefficient occurs in the set of equations by giving.
Birne Binegar: Birth index tells you which equation it lives in, and the second index tells you which variable it's tied to right, so you notice you'd look scan vertically, the second index is not changing nor is the second variable changing.
Birne Binegar: Right that's going to be our standard way of writing equations.
Birne Binegar: We have coefficients we have variable.
Birne Binegar: And we have numbers that appear on the right hand side that I will tend to call in homogeneous terms for the system of equations.
Birne Binegar: But on the right hand side and just both to have numbers now equation one i've written the whole system out symbolically I haven't told you what number a one one is but, in any particular example i'll replace the Ai jays i'll replace them with particular numbers.
Birne Binegar: So.
Birne Binegar: Let me phrase it this way, a generic two by two system.
Birne Binegar: This would look like a one one X one plus 812 X two is even be one person equation, then a two one X X one, but a two two X two X equal beat you.
Birne Binegar: or a specific.
Birne Binegar: Two by two system.
Birne Binegar: That might be two X one plus.
Birne Binegar: three x two X equals one and X one minus X equals three so.
Birne Binegar: This allows me to talk about systems in general, but when I want to look at a particular.
Birne Binegar: equation to solve that i'll replace the coefficients and the numbers on the right hand side with particular numbers The object is always going to be to figure out the correct values for the variables so set the equation is satisfying.
Birne Binegar: Alright, so.
Birne Binegar: Eventually i'm going to prove this theorem but i'm going to state it initially just So you see.
Birne Binegar: The relationship between.
Birne Binegar: The first topic, which was hyper planes and the systems of linear equation my statement is is that anytime you solve.
Noah Hightower: A linear system like.
Birne Binegar: This it's solution set is either going to be empty meaning there's no solutions or it's going to be a hyper plane.
Birne Binegar: Now, what does it mean to have an empty solution.
Birne Binegar: give you an example.
Birne Binegar: suppose I have X one X two is equal to one X one minus X two is equal to zero or sorry X one plus x is equal to zero so here, I have two equations.
Birne Binegar: With actor linear equations.
Birne Binegar: In to announce.
Birne Binegar: That will constitute a two by two linear system but there's no solution, because you cannot simultaneously have X one.
Birne Binegar: plus x to equal one, while at the same time having X one plus x two X equals zero because transitive it of equality means that one is going to have that equals zero so the set of equations implies that one is equal to zero, which is false.
Birne Binegar: So.
Birne Binegar: it's mathematical contradiction.
Birne Binegar: So solution sets like this are not going to have us equation sets like this or not gonna have any solution.
Birne Binegar: My statement here if I simply randomly write down list numbers time variables and set them equal the other numbers, I cannot always expect solutions justice in this example shows.
Birne Binegar: But so long as the equations make sense, then my statement of the serum is that when we actually saw the linear system will find that the set of.
Birne Binegar: variables X one X in right them as directors, you know the value of X one one number all right that Jason to that the value of X to to be another number, and when I write them all down and i'll have an ordered list of numbers.
Birne Binegar: Instead of all possible solutions, then it's going to be a set of factors in my claim is is always going to be a hyper plane it won't be anything more exotic than that.
Birne Binegar: So.
Birne Binegar: The proof of this theorem.
Birne Binegar: Well that's coming you know we will we will see this theorem as soon as we developed a technique for explicitly solving linear systems of this form.
Birne Binegar: But.
Birne Binegar: And in fact that's the next topic well when leading up to the next topic which is how this all those linear system, but before we actually solve them i'm going to.
Birne Binegar: make a break from what you learned in high school about solving equations and we're going to reformulate it in a purely math 3013 linear algebra way.
Birne Binegar: To do so, the first step is going to be to gather the data that goes into specifying a linear system and organizing it a little bit better so here's a typical n by n linear system.
Birne Binegar: Any equations.
Birne Binegar: In eminem's.
Birne Binegar: What we'll do is we're going to rip this apart into three pieces, first of all, the numbers that are multiplying the variables in each equation, those are the coefficients.
Birne Binegar: we're going to collect those together the call them what into an object that will call the matrix we're also going to take the barriers.
Birne Binegar: remove them from the equation you create another linear algebra a country called the matrix and what we have to take the numbers.
Birne Binegar: On the left hand side i'm going to consolidate them into a linear algebra object, called the matrix all which raises the question, what is the matrix well here we go.
Birne Binegar: A matrix is an arrangement well and N by n matrix is an arrangement of n times n numbers into an order rectangular Ray would n rows and cold.
Birne Binegar: Okay, so here, you see this rectangle arrangement of numbers, a one through eight and m how many numbers are in this arrangement well they're going to be in rows and i'm going to have.
Birne Binegar: In each row you see i've got exactly am numbers and so what i've gotten down here is a rectangular arrangement of a total well.
Birne Binegar: Each row has m numbers so i'm going to have n times n numbers told these coefficients AJ until there you go that a matrix.
Birne Binegar: And there's an ordering but it's sort of naturally will always use sub scripts to indicate were given coefficient lives inside its matrix.
Birne Binegar: notice that the entry in the.
Birne Binegar: Last column of the second row has indices to NM the first index indicate which row it lives in and the last index that last subscript indicate.
Birne Binegar: Which column is living in.
Birne Binegar: All right, then.
Birne Binegar: So what we are going to arrange these coefficients that occur in a given set of equations we're going to consolidate them into this n by n matrix we're going to take the numbers on the right hand side.
Birne Binegar: and effectively we're going to put square braces around them, you really write them in the same order from top to bottom and that's going to be a matrix with n rose, we have an slots to fill in here, but a single call, making it an end by one matrix.
Birne Binegar: And that's usually referred to that this particular construction, we have a single call we call that an n dimensional column vector.
Birne Binegar: And I could also decide to that might have a matrix with a single row, but many columns, in which case i'll refer to that as an n dimensional rogue actor, if I have an entries in a given row.
Birne Binegar: So what you observe here is that there's actually two different ways of writing vectors it may trustees.
Birne Binegar: To be a vector I just need a list of numbers peering in a certain order now here i've written be one to be in i've written them down.
Birne Binegar: spacing them vertically, but still the order is clear be one is first be to a second into the end is the last element certainly.
Birne Binegar: In this role vector here i've got an element now written horizontally, but they're ordering is clear i'm reading from left to right as usual and English now now, this is the first component, followed by the second component, so the ordering is also clear there.
Birne Binegar: Yes, now what i'm going to suddenly doing here is i'm changing my notion of vectors effectively.
Birne Binegar: you've already seen this in the Web assign problems where they tend to write vectors vertically where you're going to do so also in this course.
Birne Binegar: So, usually, when I talk about a vector i'm going to tend to think of it as an n dimensional palm nectar because then i'll be more easily related to other matrices.
Birne Binegar: Alright, so now.
Birne Binegar: you're right explain the differences between called vectors and row vectors you can think of a vector as an order list of numbers as being capable of being represented, either as a column vector or zero vector.
Birne Binegar: But when you think of it as a matrix there's only one matrix there's only one in by a matrix that represents this particular matrix.
Birne Binegar: Okay, so one other thing before we move on.
Birne Binegar: There are some.
Birne Binegar: location conventions.
Birne Binegar: When I think of a DJ is.
Birne Binegar: An element.
Birne Binegar: Of a matrix.
Birne Binegar: capital at DJ indices on the subscription can indicate the position which this element sits inside this rectangular array.
Birne Binegar: What you'll have is the first index is always going to tell you which row it lives in, and the second index is always can tell you which call lives in.
Birne Binegar: la we are here to that Convention throughout the course.
Birne Binegar: Alright, so.
Birne Binegar: here's how we use major cities to specify in by in linear system.
Birne Binegar: We take the variables that we have an invite in linear system that means a system and linear equations in an unknown we arrange the unknown vertically in a call.
Birne Binegar: We take the the.
Birne Binegar: you're on when i'm setting up here is the matrix representation of this linear system in one one X one plus.
Birne Binegar: A one two X two plus similar terms until you reach a one mm X m is equal to being one that'd be the first equation, you know, had similar equations until I reached the end the equation.
In.
Birne Binegar: X is equal to be, in effect, so this explicitly is any equations in eminem's what i'm doing on this slide here is i'm.
Birne Binegar: Taking all the variables X one X am you know i'm writing them as a column vector.
Birne Binegar: Then i'm taking all the coefficient a one 1am you know i'm i'm arranging him in this rectangular re n n by n matrix and, lastly, what i'm going to do is i'm going to take these numbers that appear on the right hand side.
Birne Binegar: Right them in the same order and then close them in braces to indicate a n by one calm enter.
Birne Binegar: So the system of equations.
Birne Binegar: happen now, the recap as a quote unquote matrix equation a coefficient matrix kinds of column vector variables equals a calling back Europe numbers.
Birne Binegar: All right.
Birne Binegar: All managed to do so far is take the data that goes into specifying writing down explicitly a system of equations and writing it down in a different way way that separates the function.
Birne Binegar: Of the various components these guys are all the coefficients, these are all variables, these are all the numbers on right hand side.
Birne Binegar: what's going to make this.
Birne Binegar: useful to us is first of all, it allows us to write a shorthand.
Birne Binegar: For the equation said, instead of having to write down all any equations and all in variables, I can.
Birne Binegar: As long as I understand at this matrix X to be this column vector and be to be this column vector I can write this entire set of equations.
Birne Binegar: In a way, that mimics what we would do for one equation in one unknown and separate now it's a matrix equation, because a reference majors not not in a number.
Birne Binegar: A second thing that's going to come out a little bit later on today's lecture is that we're gonna eventually interpret the right hand side as being the product of a matrix a with.
Birne Binegar: column vector X it's a kind of multiplication operation has been performed here.
Birne Binegar: And once we understand what that operation is that in fact this matrix equation is going to be exactly the same as this original linear system it's not going to just be an occasional place.
Birne Binegar: For separating coefficients from variables from numbers on the right hand side it's actually going to be other gentlemen equation that we compare terms on one side directly with terms on the opposite side.
Birne Binegar: Alright, but first let's.
Birne Binegar: Make sure we understand exactly how we convert from equations into this matrix notation so here's three equations in a total of four unknowns.
Birne Binegar: And what i'll do is i'll first extract the coefficient matrix alright, so the question that should be on your mind right now is how did I get this matrix from this set of equations look at that first equation X one.
Birne Binegar: plus two X three plus x four is equal to one, let me write the left hand side and a little bit more.
Birne Binegar: You know more verbose fashion X one times one that's the same thing as X one plus zero times X to that it's not going to contribute, often the sun.
Birne Binegar: But two times X three plus one times X four equals one alright, so now I explicitly inserted all the variables into the first equation and they're occurring with certain coefficient but it's.
Birne Binegar: This right, left hand side here is, can we who exactly the sun so it's the same.
Birne Binegar: But.
Birne Binegar: running out and sort of long form here, I now can identify the coefficient of each variable one for X 104 X two.
Birne Binegar: X three in one four X four and those numbers in the same order are exactly the first row of this coefficient matrix.
Birne Binegar: Second, one is two times X one minus one times X, to the other two variables extreme next for don't occur, so we say they occur zero coefficient.
Birne Binegar: And I don't leave these slots empty I put zeros there, even if the extra next for don't occur in the equations.
Birne Binegar: And the third equation, the first variable doesn't occur, and so in the coefficient matrix I put a zero in there that's zero simply means X one doesn't occur in the corresponding equation, this one means that in the last equation, the variable X two occurs with coefficient one.
Birne Binegar: All right, and then the in homogeneous term that's even that's easy you just read off the numbers on the right hand side of the equation write them down vertically, in the same order and then close it with square brackets.
Birne Binegar: Alright, so the matrix equation, the equivalent to the original system of for linear equations in three unknown is this matrix equation.
Birne Binegar: Right now.
Birne Binegar: At this stage of the game i'm making things more complicated i'm taking something that you recognize.
Birne Binegar: As a system of equations in expressing is something unfamiliar a matrix equation what we'll do next is tell you how do we interpret the left hand side here is a product that can be evaluated and compare with the right hand side.
Birne Binegar: So.
Birne Binegar: yeah what what i've stated on on the previous slide yeah and it's exactly what i'm going to do next.
Birne Binegar: Time so i'm going to introduce matrix multiplication slowly, step by step, giving you some simple examples and then we'll develop the general formula for multiply an arbitrary matrix time to an arbitrary matrix.
Birne Binegar: makes your seat, not necessarily being actors and calm actors, but we'll start with the simplest case where we multiply and row vector kinds of column vector.
Birne Binegar: So here we go.
Birne Binegar: let's let boldface lowercase Latin A, B and N dimensional robe actor, but here's written out explicitly and i'll let be be n dimensional palm nectar and then i'm going to be fine this product when i'm right a row vector adjacent to a column vector.
Birne Binegar: Of.
Birne Binegar: That multiplication is going to be defined as effectively the dot product of this ordered list with this ordered list right see.
Birne Binegar: i'm writing these components, where they are right in portal horizontally or vertically they're still ordered list of numbers so they're still n dimensional vectors and I can take their doctrine.
Birne Binegar: Meaning I multiply the first component on the first component and add to that the second component kind of the second component add to that the third component time third component until I reach the last component of both factors.
Birne Binegar: Right, so when I multiply rogue actors.com vectors it just the usual dot product, the vectors it's just that my vectors are now interpreted as particular matrices you the road matrix digital COM intricacies.
Birne Binegar: With that understanding.
Birne Binegar: move on to the next example where we take up more general factor on the left hand side will pick an n by n matrix and i'll show you how you multiply yeah and m dimensional pollen vector by this n by n matrix the rule is going to be this.
Birne Binegar: Each row of this matrix can be interpreted as an n dimensional vector as can this single column So what do I do to get the components of this product, a times X is i'll pick the product of a given row of a with this.
Birne Binegar: The vector corresponding to this call so first component controls component it's going to be a one one times X one.
Birne Binegar: second component is the second component here is can be a one two times the second column here will be X two and I keep multiplying.
Birne Binegar: elements of the first row with the matrix a kind of corresponding element of this column vector and that will fill out the first component of the matrix product as.
Birne Binegar: The second row it's going to be formed by taking the second rule a and thinking it's dot product with.
Birne Binegar: The column vector X and I do that for all all the rows of pain.
Birne Binegar: So what happened here is, we took in by a matrix.
Birne Binegar: And i'm multiplying.
Birne Binegar: A call nectar entries when a single Paul and what's getting kicked out of this is a matrix with entries in a single call so notice this result.
Birne Binegar: i'm adding these numbers together to get a single number, so this first component of the product is going to be a single number.
Birne Binegar: The second component of the product is going to be a single number so Union this product, a X is going to be a vertical arrangement of a total of in numbers so it's going to be in by one matrix.
Birne Binegar: Okay.
Birne Binegar: So here's that same formula is written out in a different way, if you want to get the Ai component of the product of an n by n matrix time to column vector it's going to be the I grew up the matrix dotted with that column vector thought of this simply as an ordered list of numbers.
Birne Binegar: Okay.
Birne Binegar: Oh here let's do an explicit example you're already having a matrix with three rows in for call until a three by four matrix multiplying a matrix four rows and a single call so a four by one matrix.
Birne Binegar: or right these guys adjacent to one another.
Birne Binegar: Which is to be understood, now as the product of this matrix times this column picture.
Birne Binegar: let's computer well what I do to get that first component of the product is, I take the Doc product of this first row with this first column.
Birne Binegar: like Britain these guys is ordered list of numbers, then the second entry of the product it's going to be the second row times the same column vector.
Birne Binegar: Third entry is going to be the third row guarded with the third entry you're right explicitly compute these products.
Birne Binegar: The first one is one time to X one plus zero times X two plus two times X three plus one times X four or simply one plus two X three plus x for you similarly compute.
Birne Binegar: The last two components and so you see that the product of these two matrices is the matrix.
Birne Binegar: And it's a new rule that let's see three by four matrix the matrix through three rows and columns is multiplying a matrix with four rows and a single home in whether it's kicking out as a result is a matrix with three rows.
Birne Binegar: And a single call.
Birne Binegar: hi so.
Birne Binegar: Now i'm in a position where I can.
Birne Binegar: Well, let me give you the point of this, so this matrix equation here.
Birne Binegar: When I write it out component by component it actually replicates the original in your system.
Birne Binegar: Okay, so, in fact.
Birne Binegar: I guess what i'm saying here is that.
Birne Binegar: If you have 1021 and I hope this was showing up 10212 to minus 100011 minus one times X one X two X three x four and you try to relate that to one free minus two what happens.
Birne Binegar: As soon as you evaluate this.
Birne Binegar: Using on rules for matrix products, what you get is one column vector with components.
Birne Binegar: X one plus two X three.
Birne Binegar: X four that's the dark product of this row with this column vector second component is going to be two X two minus X three and the third component is going to be X two.
Birne Binegar: plus x three minus X four and then the extreme side of this matrix equation, are going to replicate the original linear system X one plus two X three x four is equal to one etc.
Birne Binegar: So in this way or matrix location is going to allow us to rewrite.
Birne Binegar: A given linear system as a purely as a matrix equation, and in fact the reason for doing so is that rather focusing on variables and substitutions to solve this linear system we're going to develop matrix techniques for solving this linear system.
Birne Binegar: So what i'm suddenly doing is reformulating this problem with the equations into a problem about a single equation involving three major cities.
Birne Binegar: Right, but to do so and, in fact, to speak more generally about matrix multiplication Let me give you.
Birne Binegar: The general definition of what it means to multiply two matrices together so here i'm going to be talking about matrices of arbitrary size, not necessarily calling vectors road actors to let a be a matrix with in rows and columns let bb a matrix with s rows and columns.
Birne Binegar: So the first statement is is that unless the number of columns of the first factor is equal to the number of rows of a second factor, the matrix product is not defined does not exist as a mathematical object.
Birne Binegar: Okay, so you can't multiply every matrix tied to any other matrix theory restrictions on when it's actually going to be a well defined object.
Birne Binegar: And here's the restriction, we need the number of rows yeah sorry, the number of columns of the first factor, the equal number of rows of the second factor.
Birne Binegar: But if that's true, then the way we get the matrix product is, we take the rose up first actor that's going to be a since we have em columns each row is going to have exactly in numbers in it.
Birne Binegar: And what we'll do is we'll take one of the rows and will take its Doc product with the a column of the second factor.
Birne Binegar: Now, how many components are going to be in the second in these columns well the matrix vs both.
Birne Binegar: Both to have as columns which is equal to the number of rows sorry s rose, which is equal to the number of columns of the first factor so i'm just going to have exactly em columns and.
Birne Binegar: This matrix be as well, and so the roads have a have em entries in the columns of be entries to I have to n dimensional vectors for which there dot product is defined.
Birne Binegar: And so that's how get.
Birne Binegar: The entries of this product so here i've written it out two different ones here's the Doc product here, you see.
Birne Binegar: In the second line here i'm keeping the row index the same for all the coefficient and i'm keep keeping the column index the same, so I am J is not changing.
Birne Binegar: What is changing, or the internal indices one one here to to their name in there, those are telling you the first factor telling you it's i'm combining.
Birne Binegar: elements in a particular column a i'm combining with a similar element, but now you know particular row of be sorry i've got that feeling slightly offset and taking elements in the rows of a i'm combining them elements of the columns of The be.
Birne Binegar: All right, let's see.
Birne Binegar: here's an easier statement of it it's definitely the same formulas are written down before.
Birne Binegar: and
Birne Binegar: i've written down again, but what I want you to observe is that these.
Birne Binegar: indices, I am J, they have a fixed range.
Birne Binegar: i'm is a is an n by n matrix it has exactly and rose until these index I the row index of these entries it a okay there, they have to be numbers between one and and I get.
Birne Binegar: A sorry this upper summation should have been should have been in.
Birne Binegar: One written down incorrectly.
Birne Binegar: should be yes, a B is.
Birne Binegar: summation is entry in the iPhone J Cole it's going to be a song from K is equal one two m.
Birne Binegar: A.
Birne Binegar: R K.
Birne Binegar: A sorry be okay Jenny so the internal index are the same.
Birne Binegar: And we saw over this.
Birne Binegar: Recently, some Okay, but the outer indexes I am change they're not changing.
Birne Binegar: But in the end, what happens is when you will evaluate all these for each of the allow values of iron Jay is you take this product, and you get something where I can range between one and and Jay can run range between one in peace, so I get it in by P matrix.
Birne Binegar: Let me do a quick example before I let you go here's two symbolic matrices.
Birne Binegar: The way i'll get the first component of the product, a B is, I take the first role of a and I think it's dot product would be to that would be a time he plus B times.
Birne Binegar: Plus B times G boy go across a row and down the columns of be the second component.
Birne Binegar: So i'm going to take the first row of a and i'm going to take it's Doc product with the second column of be so that's going to be a times X plus B times ah that's right got written here and there, on the remaining component matrix.
Birne Binegar: For this product A and B is this going to be this two by two matrix.
Birne Binegar: Okay.
Birne Binegar: i'll go through many more examples of matrix multiplication.
Birne Binegar: Now, on Friday, if let's see, I have a question Well, the answer to matrix multiplication equal the rows of the first column, and the column of the second every time, yes.
Birne Binegar: that's my statement here the kind of matrix that's going to be kicked out when you multiply n by n matrix times in by P matrix it's going to be a matrix with the same number of roses, the first factor in the same number of columns at the second factor.
Birne Binegar: Okay, so that the matrix product includes in his definition the kind of matrix it's gonna pop out when you multiply two matrices together.
Birne Binegar: That i'm Okay, so I finished like sure if there are more questions i'll hang around until questions stuff Okay, but you're free to go with this point.