WEBVTT mathematics/linear-algebra/hovasapian
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Welcome back to linear algebra, so we have talked about linear systems, we have talked about matrix addition, we have talked about scalar multiplication, things like transpose, diagonal matrices.
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Now we are going to talk about dot product and matrix multiplication, so matrix multiplication is not numerical multiplication, yes it does involve not just standard multiplying of numbers, but it's handled differently.
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And one of the things that you are going to notice about this is that matrix multiplication does not commute.
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In other words, I know that 5 times 6 = 6 times 5, I can do the multiplication in any order and it ends up being 30.
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However if I take the matrix A and multiply by a matrix B, it's actually not the same as the matrix B multiplied by A.
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It might be, but there is no guarantee that it will be, and in fact most of the time it won't be, so that's the one thing that's actually different about matrices and then numbers.
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Le's just jump in, get started and see what we can do, okay.
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Let's go ahead and start with a definition and this is going to be our definition of dot product, which is going to be very important and it shows up in all areas of science., so we will let...
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... equal, A1, A2, A3, let me erase this A3 and just use ... all the way to AN.
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And B = B1, B2..... BN, okay so let me just explain what these notations means here.
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Whenever we see a normally a lowercase letters a, b, c, d, x we will often use x with an arrow on top, that means it's a vector, and a vector is just a list of numbers.
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A1, A2 all the way to AN in this particular case we are talking about an N vector, which means it has N entries, so 5 vector would have 5 entries.
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An example might be, let's say the vector V might be 1, 3, 7, 6, that's all this means, this is the vector in these the components of that vector.
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It's composed of (1, 3, 7, 6), it's a four vector, because it has four entries in it, that's all this notation means, this is just a generalized version of it.
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Okay, so let A, the vector A = A1 to An, let the vector B = B1 through BN, now we defines something called the dot product as the following, A.B.
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The product of two vectors is equal to A1 times B1 + A2 times B2 + ... +A < font size="-6" > n < /font > B < font size="-6" > n < /font > and I am going to write this in σ notation.
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σ notation, I'll explain in just a minute, if you guys haven't seen it, I am sure you have, but you just, I know that you don't deal with it all too often.
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Okay, so if the vector A is composed of A₁ through AN, B is the list, B₁ through BN, the dot product A.B = the product of the corresponding entries added together.
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When I add these together, I end up with a number, so the dot product of two vectors gives me a scalar; it gives me a number, so I just add them all up.
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This σ notation is the capital Greek letter S, and stands for sum, and it says take the sum of the Ith entry of A, the Ith entry of B.
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Multiply them together and add them, so A1B1, 1I = 1, and then go to the next one, I = 2 + A2B2 abd then go to I = 3 + A3B3.
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This is just a short hand notation for this, we won't deal with σ notation all that much, what end our definitions, whatever we do I'll usually write this explicitly.
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I just want you to be aware that in your book, you'll probably see this; you'll definitely see it in the future.
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That's all this means, it's a short hand notation for a very long sum, so don't let the symbolism intimidate you, scare you, confuse you, anything like that, it's very simple.
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Okay, let's just do an example of a dot product and everything should make sense, so example; we will let...
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... Vector A = (1, 2, -3 and 4), so this is a four vector, and will let B = (-2, 3, 2, 1), notice I wrote one of them in row form, one of them in column form.
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This is also a four vector because we have four entries, I wrote it this way because in a minute when we talk about matrix multiplication, it's going to make sense, it will make sense why it is that I wrote it this way, but just for now understand that there is no real difference between these two.
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I could have written this as a column, I could have written this as a row, it's just a question of corresponding entries.
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But I did like this because in a minute when we do matrix multiplication, symbolically, its going to help make sense when you move your fingers across a row and down a column, just sort of keep things straight, because matrix multiplication, there is lot, a of lot of arithmetic involved.
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Okay, so our dot product A.B here, A.B, we just go back to our definition, it says take corresponding entries and just multiply it together, that's you got to do,
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I take A1 times B1, so which is 1 times -2, which is -2 + 2 times 3, which is 6 + -3 times 2, which is -6...
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... + 4 times 1, which is 4.
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Well, that equals, so the 6 is cancelled, -2 + 4 gives me a 2, so I have a vector of 4 vector, times of 4 vector and I end up with a number 2.
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The dot product of two vectors is a scalar, and all I am doing is multiplying corresponding entries and adding them all up, that's it simple arithmetic, nice and easy, no worries.
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Let's go ahead and move forward now on to matrix multiplication, okay.
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Let me go ahead and write down the definition of matrix multiplication and then we wi do some examples.
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We will let A = that matrix A < font size="-6" > IJ < /font > , B...
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... M by P, so this is an M by P, be 3 by 2.
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Let B be the IJth P...
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P by N, so A is M by P and B is P by N, notice that the number of columns of the matrix A equal to the number of rows of the matrix B, that's going to be very important.
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Then AB is the...
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... M by N matrix.
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C equals C < font size="-6" > IJ < /font > , such that the IJth entry of C is equal to Row < font size="-6" > I < /font > of A.
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In other words, the Ith row of A dotted with the Jth column of B.
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let's take a look at the definition again, A is a matrix, A < font size="-6" > IJ < /font > , it is M by P, B is a matrix, B < font size="-6" > IJ < /font > is P by N.
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when I multiply those two matrices the, essentially what happens is that the column of the first matrix, the one on the left cancels the column accounts with the row of the matrix on the right an what you end up with is a matrix which is M by N.
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And that matrix is such that the IJth entry = Ith of A dot end with the Jth column of B, that's why this P and this P have to be the same.
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In order to multiply two matrices, let's write this one out specifically, okay.
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In order to multiply two matrices....
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... The number of columns of the first...
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... Must equal...
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... The number of rows of the second and that's what this says M by P, P by N, the number of columns of the first has to equal the number of rows.
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That's the only way that matrix multiplication is defined and what we mean when we say is defined, means if they are not the same, you can't do the multiplication.
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That's what defined means, it's the only way you can do it if that's the case, okay.
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let's see what we have got, so for example if I have a 2 by 3 matrix and I want to multiply it by a 3 by 2 matrix, yes I can do that because the number of columns of the first one is equal to the number of columns of the second one, and essentially they go away.
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What I am left with is the final matrix which is 2 by 2, this is kind of interesting.
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Now notice if I reverse them and if I did a 3 by 2 matrix, and if I multiply that by a 3 by 2 matrix, I am sorry 2 by 3...
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... Now, it is defined, number of columns of the first equals the number of rows of the second, so now I end up with a 3 by 3 matrix, okay.
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These are all defined, that will work.
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Let's see 2 by 3, 3 by 2, 3 y 2, 2 by 3, so notice what's happened here, take a quick look, I have a 2 by 3 times the 3 by 2 gives me a 2 by 2.
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If I switch these, a 3 by 2 by a 2 by 3 I'll let them switch them, I get a 3 by 3, a 3 by 3 and a 2 by 2 are not the same.
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In general not only do the dimensions not match, it won’t work, AB is not equal to BA, that’s the take on lesson for this, matrix multiplication does not commute.
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AB does not equal BA, and we will actually do an example later on where we can actually do AB and BA, but they end up being completely different matrices.
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Okay, let's do some examples, let's let A = (1, 2, -1, 3, 1, 4) when doing much matrix multiplication goes very slowly and go systematically.
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Lot of arithmetic, lots of room for mistake, (-2, 5, 4, -3, 2, 1) okay.
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We said that the IJth entry = Ith row of A times the Jth column of B, well we are looking at here, A, let me use black, this is a 2 by 3, 2 by 3 matrix and this is a 3 by 2.
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Yes, it is defined because this 3 and this 3 are the same, so we should end up with a 2 by 2 matrix, okay.
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Lets go ahead and put little thing here for our 2 by 2 matrix, now for our...
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.. This, the first row first column, this entry it's going to equal the first row of A dotted with the first column of B, so it's going to be that row and that column, so I take 1 times -2.
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Let me actually write over here or let's call this the, so now we are doing the A11 entry, this one right up here, A11 entry equals 1 times -2, which is -2, 2 times 4 which is 8, -1 times 2, -2.
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-2 - 2 + 8, answer should be 4, so 4 goes there.
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Now let's do this entry which is the first row, second column, well the first row, second column means I take the dot product of the first row of A and the second column.
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A12 Entry = 1 times 5, which is 5, 2 times -3, which is -6, -1 times 1, which is -1.
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That means, let's try this again without these little extra lines, 1 times 5 is 5, 2 times -3 is -6, -1 times 1, -1.
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5 - 6 is -1, -2, so this becomes -2, now we are going to go to the second row, first column, which means we do second row first column.
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This is A21, 3 times -2 is -6, 1 times 4 is 4, 4 times 2 is 8, so 8 + 4 is 12 - 6 is 6, so this entry is 6.
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And now we have our last entry which is the 2,2, so the 2,2 entry, second row, second column, which means we dot product the second row with the second column, second row of A, second column of B.
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3 times 5 is 15, 1 times -3 is -3, 4 times, oops, that's nice.
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4 times, is that a -1, I don’t even know, no that's 1...
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4 times 1, is 4, so we get 15 + 4,., which is 19, 19 - 2 is 16, so this entry is 6.
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The product so, AB = 4 - 2, 6, 16, 2 by 3 matrix multiplied by a 3 by 2 matrix gives us a 2 by 2 matrix, and we get that by this row this column, this row this column, and then this row this column, this row this column.
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That's all you are doing, rows and columns, now you know why I arranged it, remember a little bit back when we did dot product, I arranged it, the first one horizontally and the other one vertically.
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This is the reason why, because when we multiply, we are doing this times that, this times that, this times that, we can move one this way, one this way, it seems sort of, it's a way to keep things separate, as one hand, one finger moves across a row.
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The other finger should move down a column as used to going this way or this way., okay.
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Lets do another example here, we will...
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... Okay, let A equal, this is going to be a 3 by 3, so it's 1(1, -2, 3, 4, 2, 1, 0, 1, -2)
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And B is equal to, let's make it a 3 by 2, so this is a 3 by 3, and then we have (1, 4, 3, -1, -2, and 2) , so this is a 3 by 2, so AB.
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A times be is defined, so AB is defined and it's going to be well, 3 goes away, the 2 inside 1's, so we are left with a 3 by 2, so AB is a 3 by 2 matrix.
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Okay, well let's just multiply it out, this time we are not going to write everything out we are just going to do the multiplication and keep it straight, they are final numbers.
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We know that we are looking for a 3 by 2, so let's just start putting in entries, well the first entry; first row first column is going to be first row first column.
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1 times 1 is 1, -2 times so 1 times, that's 1, -6 is going to be a -5, and then -6 is going to be -11.
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This is going to be -11, there and now we are going to do the second entry, okay, first row second column, which means first row of A, second column.
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One times 4 is 4, -2 times -1 is 2, 4 + 2 is 6, and then 3 times 2 is 6, that becomes 12.
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When we continue ion this way, we end up with 8, we end up with 16, we end up with 7, we end up with -5, that's our AB.
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Okay, now let's try something.
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Let's let A =(1, 2, -1, 3) and we will let B = (12, 1, 0, 1) in this case, because this is 2 by 2, and because this is 2 by 2, both AB and BA, they are both defined.
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I can do the multiplication, well let's do the multiplication and see if AB = BA.
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There are two ways that you can, there are certain demonstrate non-commutivity, is if the dimensions don't match when you switch them or if it's defined, multiplication is defined and doable this way and that way.
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Then you might end up with different matrices, again proving that it doesn't commute, alright.
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Let's see what we have got, when we do AB, okay we end with the following, we end up with (2, 3, -2, 2) and when we do BA, we said it is defined.
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We end up with (1, 7, -1, 3).
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AB and BA are not the same, AB is not equal to BA, matrix multiplication does not commute.
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Okay, so now let's talk about matrices and linear systems, so we introduced linear systems in our first lesson, we talked about matrices in our second, and we have just introduced matrix multiplication.
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Now let's combine them together to see if we can take a matrix and represent it as a linear system, or a linear system and represent it in matrix form.
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Let's let me go back to blue here, we will let, excuse me, A = A < font size="-6" > 11 < /font > , A < font size="-6" > 12 < /font > , A < font size="-6" > 13 < /font > , A < font size="-6" > 21 < /font > , A < font size="-6" > 22 < /font > , A < font size="-6" > 23 < /font > , A < font size="-6" > 31 < /font > , A < font size="-6" > 32 < /font > , A < font size="-6" > 33 < /font > .
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And we will let X with the little line, the vector be our vector, let's call it X₁, X₂, X₃, this is the vector formulation, this is the component form, it's just a 3 vector.
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Okay, so this is a, we can do this in red, this is a 3 by 3 matrix, and this is a 3 by 1, right, so if I multiply this matrix by that vector X, well it's just a 3 by 3 times a 3 by 1.
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Well those are the same, so I end up with a 3 by 1, it is defined and it's going to equal some vector b, which is going to be a 3 by 1 vector, just something with 3 entries in it.
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And let's let B therefore equal, we will call it B₁, B₂, B₃, so again we have a matrix.
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We have this 3 vector, I can multiply them because matrix multiplication is defined, their answer is going to be a 3 vector, so we will call that 3 vector B, and will call it's components B₁, B₂, B₃.
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Okay, well let's actually do the multiplication here, so A₁ X, I am sorry, AX.
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When i do this multiplication, this row, this column, this row, this column, this row, this column.
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Here is what I get, A < font size="-6" > 11 < /font > times X₁ + A < font size="-6" > 12 < /font > times X₂ + A < font size="-6" > 13 < /font > times X ₃, that's what I get, that's the multiplication.
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A < font size="-6" > 11 < /font > X₁ + A < font size="-6" > 12 < /font > X₂ + A < font size="-6" > 13 < /font > X₃, and then I do this second row, that column again, I get A < font size="-6" > 21 < /font > X₁ + A < font size="-6" > 22 < /font > X₂ + A₂ X₃.
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And then I will do the third row, which is A < font size="-6" > 33 < /font > X₁ + A < font size="-6" > 32 < /font > X₂ + A < font size="-6" > 33 < /font > X₃, that's going to be my matrix.
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That's my actual matrix multiplication; well I know that equals this B, so I write B₁, B₂, and B₃.
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Well, this thing = this thing, this thing = equals this thing, this thing = this thing, that's what this says, this is just a 3 by 1 in its most expanded form.
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That's the A times the X, this thing is the B, that are equals, and so now I am just going to set corresponding entries equal to each other, this whole thing is equal to that.
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I write A < font size="-6" > 11 < /font > X₁ + A < font size="-6" > 12 < /font > X₂ + A < font size="-6" > 13 < /font > X₃ = B₁.
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A < font size="-6" > 21 < /font > X₁ + A < font size="-6" > 22 < /font > X₂ + A < font size="-6" > 23 < /font > X₃ = B₂, and I am sorry that I have got extra little lines here that are showing up.
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Try to spread a little bit slower, A < font size="-6" > 31 < /font > X₁ + A < font size="-6" > 32 < /font > X₂ + A < font size="-6" > 33 < /font > X₃ = B₃.
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Well take a look at this, this is just a linear system, that's it, it's just a linear system, you have seen this before.
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This is three equations in three variables...
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...X₁, X₂, X₃, X₁, X₂, X₃, X₁, X₂, X₃, these A < font size="-6" > 11 < /font > 's A₂, all of these are coefficients and these are the actual solutions.
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You can actually write a linear system as a matrix, so it looks like A < font size="-6" > 11 < /font > , A < font size="-6" > 12 < /font > , A < font size="-6" > 13 < /font > , this is the coefficient, the matrix of coefficients for the linear system.
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A < font size="-6" > 21 < /font > , A < font size="-6" > 22 < /font > , A < font size="-6" > 23 < /font > , A < font size="-6" > 31 < /font > , A < font size="-6" > 32 < /font > , A < font size="-6" > 33 < /font > , and then you multiply it by the variables, which are X₁, X₂, X₃, and it equals B₁, B₂, B₃.
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We can take a linear system and represent it in matrix form; we take the matrix of coefficients, so this is the coefficient matrix.
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M by N in this case is 3 by 3m, but it can be anything, this is the matrix of variables, it's the variable matrix, and it's always going to be some N vector.
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And this is just the you might call the solution matrix, it's not really the solution matrix, the solution matrix is once you find X1, X2, X3, those are going to be your solutions, so you know what let's not even give this a name, let's just say this happens to be the, whatever it is.
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It's the B that makes up linear system on the right side of the equality, okay now given this; we can actually form as it turns out.
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We can form a special matrix...
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... If we attach...
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... B₁, B₂, B₃ to the coefficient matrix...
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... Okay, now we are going to write this out one more time, so I am going to take this thing and I am going to add up another column to this matrix.
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I end up with, (A < font size="-6" > 11 < /font > , A < font size="-6" > 12 < /font > , A < font size="-6" > 13 < /font > ), B₁, (A < font size="-6" > 21 < /font > , A < font size="-6" > 22 < /font > , A < font size="-6" > 23 < /font > ), B₂, (A < font size="-6" > 31 < /font > , A < font size="-6" > 32 < /font > , A < font size="-6" > 33 < /font > ) B₃.
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And sometimes we put like a little dotted line there, just to let you know that this is, and this is called an augmented matrix.
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All I have done is I have augmented my coefficient matrix with my solutions on the right for the linear system, and we do separate it.
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Sometimes you can see a solid, I tend to put a solid line, that's just my personal preference, some people don't put anything at all, again it's, it's completely up to you.
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Therefore any linear system can be represented in matrix form and vice versa, any matrix with more than one column can be thought of as forming a linear system.
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Let's see what we have here, example -2X as a system -2X + 0Y + Z = 5, and then we have 2X + 3Y -4Z =7.
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We have 3X + 2Y + 2Z = 3, so this is our linear system and now let's break it up and in matrix form, so we want to write it this way, AX = B, this is the matrix representation.
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A matrix A times a vector B gives us a vector B, so A the matrix A is going to be the coefficients, it' going to be , excuse me, the 2, the 0, the 1, the 2, the 3, the -4, the 3, the 2, the 2.
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(-2, 0, 1, 2, 3, -4, 3, 2, 2) that’s our coefficient matrix, X let me do this in red, oops.
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X is going to be our variables, our variables happen to be X, Y and Z, that's going to be X, Y, and Z, and B, let me go back to red, B vector is going to be (5, 7, 3).
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That's it, you can represent, now we will do the augmented matrix, which means take the coefficient matrix and add this to it, so we end up with A augment with B, symbolized like that.
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It is equal to (-2, 0, 1, 5) and I'll go ahead and do a solid line, because I like solid lines.
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(2, 3, -4, 3, 2, 2), you have your coefficient matrix and you have your matrix that represents the linear system, that was originally given to you like that.
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Now, let's see, now let's go the other way, let's say we have a matrix, (2, -1, 3, 4, 3, 0, 2, 5) let's say you are given this particular matrix, this particular matrix actually can represent a linear system.
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We could take a linear system, represent it in matrix form, which we just did, we can take a matrix and represent it as a linear system, if we need to.
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This ends up being, so let's say that this is the augmented matrix, so that means this is (1, 2, 3), that means we have 3 variables, that's what the column represent are the variables, and these are the equations.
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We have 2X - 1Y + 3Z = 4, and then 3X + 0Y + 2Z = 5.
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Linear system, matrix form, matrix represent a linear system.
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Excuse me,, there you go, okay now let's talk about what we did today, recap our lesson.
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We talked about the dot product of two vectors and a vector is just an N by 1 matrix, either as a column or row, it doesn't really matter.
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What you do is you multiply the corresponding entries in the two vectors and you add up the total.
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The dot product gives you a single number, a scalar, it's also called the scalar product, so dot product, scalar product, as you go on in mathematics you will actually refer to it as a scalar product not necessarily an dot product.
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After that we talked about matrix multiplication where we actually invoke the dot product, so with matrix multiplication you can only multiply two matrices if the number of columns in the first matches the number of rows in the second.
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Matrix multiplication does not commute, in other words A times B does not equal B times A in general.
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It might happen accidentally, but it's not true in general.
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The IJth entry in the product is the dot product of the Ith row of the first and the Jth column of the next.
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Okay, now matrix representations of linear systems, any linear systems of equations can be represented as an augmented matrix, you take the matrix of coefficients and you add the column of solutions.
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Any matrix with more than one column can represent a linear system of equations, that last column is going to be your solutions, that's the augment.
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Okay, so let's do one more example here, so we will let A = (3, 5, 2, 4, 9, 2, excuse me.
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And B = (1, 0, 1, 6), oh she knows, (2, 1, 3, 7) so here we have a 3 by 2 matrix and here we have a 2 by 4 matrix.
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Yes, the 2's on the inside are the same, they end p cancelling and it's going to end up giving up a 3 by 4 matrix, so we are left with 2 outside.
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We are going to be looking for a matrix which is 3 by 4, this is kind of interesting if you think about it, 3 by 2, 2by 4, now you get a 3 by 4, you get something that's bigger than both in some sense, okay.
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AB equal to, well we take the first row and first column, 3 times 1 + 5 times 2, 3 times 1 is 3, 5 times 2 is 10, you end up with 13, first row, second column.
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Well you take the first row, dot product of the second column, 3 times 0 is 0, 5 times 1 is 5, so you end you with 5., then you keep going.
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you end up (18, 53, 8, 4, 14, 40, 13, 2, 15 and 69), so our product AB = this matrix.
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Notice 8 times B is defined if I did B times A, well B times is equal to a 2 by 4 times the 3 by 2.
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This 4 and this 3 aren't equal, BA is not even defined, we can't even do the multiplication, leave alone and find out whether it equals or not, which in general it doesn't, so in this case it's not even defined.
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It only works when A and B are such that A is on the left of B, B is on the right of A, and they will often say that, we will often say in linear algebra, multiply by this on the left, multiply by this on the right.
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We don't do that with numbers, we just say multiply the numbers, okay now let's let the variable that's the X, the vector = X, Y and Z and let's let the vector Z = (4, 2, 9).
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Now we want to express...
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.. Well actually we don't want it, let's go ahead...
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... We want to express AX = Z as a linear system and as an augmenting matrix, both, so we have a matrix A, that's this one; we have a matrix X, we have a matrix Z.
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We want to express AX = Z as a linear system in augmented matrix, okay, we’ll wait a minute, let's try it to bring it, it works.
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We can't even do, this is, this has 3 and 2, so this can be XYZ, this is going to have to be XY, my apologies.
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XY, there we go, because this is 3 by 2, this is 2 by 1, yes we want them at, the multiplication to be defined, so we end up with (3, 5, 2, 4, 9, 2)...
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... Times XY = (4, 2, 9) so you end up with, well 3X + 5Y = 4.
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3X + 5Y = 4, because you are doing the first row, first entry, first row first column, 2X + 4Y, 2X + 4Y = 2.
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And 9X + 2Y, 9X + 2Y = (, all we have done is go this times that, equals that, this times that equals that, this times that equals that.
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And we end up with our linear system, now we want to convert that to an augmented matrix; well we take 3, 2, 9, (3, 2, 9, 5, 4, 2).
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I'll start coefficient matrix, right, and we have augmented with our solution matrix 4, 2, 9, (4, 2, 9).
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That's all we have done AB, A times B, it is defined, we can find the multiplication.
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We can take given X and given Z, this is a two vector, this is a three vector, we can take AXC represented as a linear system.
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We express it this way, we do the matrix multiplication, we said corresponding things equal to each other, and we have actually converted this to a linear system.
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This and this are equivalent, we can take this linear system and express it completely just as a matrix, an augmented matrix by adding the solutions as the augment on the right, and we end up with that.
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Okay, thank you for joining us today for linear algebra, and our discussion of dot products and matrix multiplication on linear systems.
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Thank you for joining us at educator.com, we will see you next time, bye, bye.