 Hi, I'm Zor. Welcome to Unizor education. I would like to introduce the operation of multiplication of matrices Well, this operation is probably the most important operation on matrices. Yes, we were talking about addition of matrices and multiplication by scalar But this is actually where the whole theory is is is really flourishing with multiplication Why well primarily because we have devised the multiplication in such a way that it actually represents the the consecutive mod the consecutive transformation of Of the space Transformation of the vectors in the space. So if you remember we have introduced Matrices as just a convenient way to to show how One particular vector is transformed into another Now let's just give you an example. For instance, we have a two-dimensional vector u Which has two coordinates u1 and u2 and we have The following transformation of the coordinates Now this is a linear transformation of the coordinates of the vector Now linear obviously because this is just a multiplication by a constant. So we are transforming vector u into V with coordinates v1 v2 according to these rules and the matrix of transformation Was actually just a table with coefficients So so far. It's nothing but symbolics Nothing but a symbolic representation of these two Equations well, obviously if our dimension is three there will be three equations, etc Now the multiplication of matrix is something which allows us to write it in the following form either this form or This form depending on certain conditions, which I will talk later so Transformation is equated with multiplication by a matrix Now that was the original purpose actually of the operation of transformation But we have to define it in certain way Which is really kind of reflecting this type of thing now more than that I would like to Make to define this transformation in such a way that if I have let's say two transformations U to be and then B to W one is represented by the matrix a and Another is represented by a matrix B Then the transformation from U to W is a multiplication a product of two matrices A and B So that's the whole purpose to build this matrix arithmetic in such a way that it reflects The processes of transformation. Well, that's the purpose question is how to achieve this purpose And that's what we will be talking about In in this and in the following lectures now this lecture is just an introduction It just to show you how the matrix actually transforms different vectors Now in the subsequent lectures, I will be spending more time defining different kinds of multiplication of matrices and And then the whole picture will become much clearer so multiplication of matrices in general is Well, it's an operation Which takes two operands two matrices producing the third operant Producing the result basically the third matrix now. I was just talking about transformation of vectors using matrices and And now I just mentioned that matrix multiplication is operations into matrices, but well look at it this way any vector can be actually Viewed as a matrix in this case it's matrix which has one row and two columns Or I can write this this way In which it's two rows and one column This by the way is called row vector and this is called column vector These are both vectors, but it's just a representation in a more matrix like format Requires us to specify exactly what kind of vector we're talking about So if I'm saying it's an n-dimensional vector and I'm not saying whether it's a row vector or a column vector Then you don't know which matrix actually represents this vector because there are two different representations Because when I'm talking about matrices, I usually have okay This is a matrix of the size m times n m rows and n columns So that's exactly what I have to specify with the vector if it's one row and n Columns, it's a n-dimensional row vector if it's n rows in one column. It's n-dimensional column vector Alright, so basically what I'm saying now is that the multiplication of the matrix by vector Which I was talking before is also Kind of matrix multiplication I'm just Considering the vector as a matrix with either one row if it's a row vector or one column if it's a column vector So generally speaking we can talk about multiplication of matrices Implying that in certain cases in certain matters for certain matrices. It's applicable to vectors as well now The first thing which I would like to do is define the multiplication of matrix by a vector so That would be actually exactly what transformation of the vectors is and that's how it started Our matrix concept well in a way. I have basically already Defined it when I was writing an equation Actually as many equations as the dimension of the space But let's do it in n-dimensional case. Let's say we have an n-dimensional vector and this case it's a row vector and Let's say we would like to transform it into another n-dimensional vector using a matrix in this case the matrix is square matrix and Rows and n columns because that's how all these equations will look like now This is the first equation and This is the last equation That should be you one so as you see the matrix has n rows and n columns and this is How it looks? Square matrix the same number of rows and columns now I can multiply it By the vector u and the vector u is this as I was saying and I'm getting the vector v Okay So this is basically a general representation of the matrix Now let's think about this in slightly shorter format What is a? Ice co-ordinate of the D Well ice co-ordinate from this it would look like this I to you to plus et cetera plus I I and You and Now what does it remind you? Well, I can tell you what it reminds me if I take If I if I take the ice row of the matrix a Ice row would be here I 1 I And consider it as a row vector So I will specify it this way I start It's a I 1 a I 2 et cetera a I n Now This is an ordered set of numbers and I can always consider it as a row vector now this is a scholar product of row vector a ice The row vector of the matrix of transformation and our original vector So I can say that this is equal to scholar multiplication of the vector row vector in the ice row of our matrix and Our original back So that's what ice co-ordinate is So the first coordinate is row Row vector in the first row scholarly multiplied by our original vector the second coordinate of the result is the second row Vector of the matrix of transformation Multiplied as a scholar product by our vector et cetera So this is an important No, which you really Should take into account because it looks simpler than something like this or something like this, right? Okay, so this is An introduction as a concept of what the product of the square matrix of the size n by n and And dimensional vector is What I will do next I will I Will explain a couple of simple cases of this particular transformation and It would be clearer to understand how Matrix really represents a transformation of the of the vector space Okay, so let's go into a couple of examples and these examples will be in the simple two-dimensional case So our vectors which we are transforming Belong to the plane So they're all two-dimensional vectors and Here is my first example. Let's take the matrix Two by two which looks like this Where k is some number doesn't matter what it is just some real number and I would like to multiply this By a vector now the vector is Any vector you want you to and see what happens? Okay, well if I multiply the square matrix by a vector I will get another vector question is what kind of this vector is well according to the equations V1 is equal to K times you one plus zero times you two a 11 a 12 a 11 a 12 Now V2 is equal to the second roll zero times you one plus K times you two So This is equal to K you one this is equal to K you two and what do we have now? We have that the vector V is equal to K Times vector U If you multiply the constant by a vector You will Multiply each component and that's exactly what the vector V actually is so the matrix of transformation Which is as stretching by the factor of K of an entire Vector space so every vector will be stretched by by by K this matrix of transformation is This particular matrix So this is an example of how the matrix Describes the transformation Algebraically so we can talk about stretching our vectors by a factor of K or we can say that all our vectors are Multiplied by this particular matrix according to the matrix multiplication Okay, now a particular case of that if K is equal to one Obviously we have no transformation, right? We multiplying by one so the matrix one zero zero one Place in the multiplication Lingo The role of unit matrix. It's a multiplication By this particular matrix results in basically no change in the vectors So this is called the unit matrix And the usual Notation is with a letter I Okay Now let's consider another example another example of a linear transformation minus one zero zero one Okay, so what kind of a transformation this particular matrix represent Well, let's multiply this by you one you two And what do we have First we have to multiply the first row by these coefficients, which is V one is equal to minus you one Plus zero times you two and we two is equal to zero times you one plus one times you two which is you two So what is this particular transformation? Well, let's just consider it this way if this was a vector you one You two it will be transformed into minus you one and You two So this is a reflection. It's a reflection relative to the y axis. It's a linear transformation, obviously and it's Represented by this particular matrix. So this matrix is a reflection So again, we can talk about reflection and we can talk about these Changes changes of the coordinate But at the same time we can say that exactly the same transformation can in a matrix format be represented by a product of this Matrix by a vector alright next example as You I'm sure Will be very comfortable with Now this would be plus this would be minus then V one would be equal to you one and the two would be equal to minus you two So what is this so you one? This is our vector you one you two so you one remains intact But you two is changing to minus you two so that would be Reflection relative to the x-axis all right That's as simple okay, another example is This What do we have here? Well, let's see V one is equal to Zero times you one plus one times you two which is you two V one V two is equal to one times you one Plus zero times you two which is equal to you one so V one is equal you two V two is equal to V two to you one So what do we have here? So this is your one. This is you two now the new coordinates The obsessive would be equal to the ordinate which is this one this is V one and The ordinate is equal to obsessa this is V two Well, obviously these are symmetrical relative to the bisector of the angle So we are changing x to y and y to x. That's what basically this is we're exchanging the coordinates and in the two-dimensional case, this is a symmetry reflection relative to the Bisector of the main angle and the last example I am boring from the lecture which Was dedicated to rotation of the vectors so If you don't remember go back to that lecture, but basically what it says is we are talking about this particular Equations So when I was talking about a rotation of the vector by the angle phi this is These are the equations which relate new coordinates with an old coordinates where phi obviously is the angle of rotation Now can I specify it in a matrix format? well, obviously, yes, I can and the specification is This is the matrix Cosine phi minus sine phi sine phi Cosine phi So if I multiply this matrix by this vector, I will get this vector, right? So this matrix specifies the rotation. So basically look what we have done right now We have specified stretching we have specified reflection different reflection and we have specified rotation basically any kind of Linear transformation is the combination of these basic transformations so Right now. I think I have completely Described how to multiply matrix by a vector Now this is just the beginning and this is related to transformation Next thing which I would like actually to show is when I specify a couple of transformations Consecutive so from one vector we go using one matrix of transportation to another vector and then from that vector Using another matrix of transportation yet yet another one the third vector question is whether my matrices Can be combined in some way. So let me just jump forward. So if I do this With this is my first transformation of the vector into let's say D and then I have another Matrix which multiply which is multiplied by this V and I get W That question is can I specify it differently? Can I define my matrix multiplication? So at first I multiply the matrices first just by by themselves and then I apply the resulting matrix To my original vector and I get whatever I get the first time now what this requires now So far I have defined only the multiplication of matrix by a vector and this is another matrix by the vector again This is multiplication of matrices and this is the subject of the next lecture All right, that's it for today. Thank you very much I would like to point to unisor.com as the source not only for these lectures But also for many different problems Solved on this and also there are some exams for registered students. I also suggest you to get Some supervision involved parents or teachers whoever Who can enroll students into this or that particular topic? check the exam and they can Mark certain topics as completed and enroll into next topics, etc. Which gives the supervisor a control over the educational process So basically that's it for today. Thanks a lot and good luck