 So today in the algebra very simple concept, and that's the transpose of a matrix Transpose of a matrix. So if I have a matrix a and it has columns i and j the transpose of that the transpose of This we usually write like that and that is this where we interchange rows of columns So if I were to have this a 11 oops, let's do a A 11 a 12 a 21 a 22 that's my matrix a So I'm going to interchange a rows of columns So a transpose is then going to be a 11 stays where it is because that changes around a 1 2 Becomes a 2 1 so it is going to move there So I'm going to have a 1 2 is now suddenly going to be there a 2 2 is still the same Rows and columns, but this one here is going to go up there So as these just grow bigger, that's what you're going to have So let's just look at the transpose of a matrix say for instance 3 2 1 4 2 minus 3 So that is a 2 by 3 matrix and you can well imagine that the transpose of that has got to be a 3 by 2 Matrix so I'm just going to take this row and make it a column So this becomes 3 2 1 and the second row becomes the second column 4 2 minus 3 and then behold this is a 3 by 2 matrix So you see that's quite simple this idea of just taking a matrix taking its transpose We just making columns out of rows or rows out of columns So what happens if I have the following I have a plus b and I take its transpose So remember they if I want to add two matrices to each other They better have the same the same side So let's make this an m by n matrix and I add to this b and this must also be an m by n matrix and The result of that will might be another matrix C But it'll also be m times n now. What happens if I take the transpose What happens if I take the transpose of This if I take C transpose that it's going to be n times n matrix Those are going to swap around the rows and columns are going to be swapped around So this was a b I added them first and then I got the transpose if I have a transpose though That's going to be n times n and if I have b transpose That's also going to be n by m and if I add them together. I'm also going to get an m an m by n So it doesn't matter what you do. This is going to be the exactly the same as a transpose plus b transpose There's going to be no difference between those where it gets tricky. Of course is if you do multiplication We we do multiplication and an easy way just to remember that is remember is remember that if we had a b inverse That was equal to b inverse a inverse same here that the transpose of a b multiplied first that's b transpose times a transpose and This is look at the dimensionality of this to see that it makes sense So imagine a is I'm going to make it a times b and Then that means b must be b times b must be b times c Because those have to be the same and this is going to give me a resultant matrix Which is a times c and if I get the transpose of that that's going to be c times a Now let's get b transpose b transpose here is going to be c times b and And ace transpose Ace transpose is going to be b times a B times a and if I multiply that out that's going to give me another matrix c Which is going to be c times a which is exactly Which is exactly what we have there. So if I take its transpose in in other words So so that's the transposing the transpose of those two So I get exactly the same thing as I as I got here So please remember that a times b transposes b transpose times a transpose so Let's have a look at one more thing which naturally Follows from this and those are the symmetric matrices So symmetric matrix is where this the matrix is directly equal to its transpose It's directly into to its transpose. Let's have one two two and Five would that work as a what would a transpose b? one and two and two and five and I have exactly the same thing I Have exactly the same thing that the matrix and its transpose are the same and those are referred to as As symmetric matrices if I do those two I get exactly the I get exactly the same thing So that is something that one and two that Joe becomes a column and that Joe becomes a column And they exactly the same thing so that will give me a symmetric matrix What is the inverse of a symmetric matrix is that also so let's stick with a symmetric matrix. I have a equals What was it one two two and five? Let's get its inverse very quickly and there are a variety of ways of doing that I showed you how to do that if we have one two two and five one and zero zero one and we did Elementary row operations, but I was going to end up with is five and one and minus two and minus two Just check on that. That's exactly what you were going to get I can do that easily in my head because with two by two matrices We just swap these two around and we multiply that by negative one and that one by negative one and we get exactly that And what is a inverse? What is the transpose of? a inverse well that row becomes a column and that row becomes a column and We see that they exactly the same so if a is symmetric its inverse is also symmetric in other words It's transpose is exactly the same as where we start So these are all very simple things and I don't think you will ever ever struggle with that It's be very easy to do one last thing perhaps is Remember vectors and vector multiplication So imagine I remember I had a vector u and it's a column vector 1 u 2 u 3 So this would be in three dimensional space and if I have v and it's v 1 v 2 and v 3 Also in three dimensional space and I want to get the inner product the dot product and I would say u dot v those two vectors u dot v and That would be u 1 v 1 plus u 2 v 2 plus u 3 v 3 And that will give me a scalar and it gives us a very nice way to do this this idea of a transpose Gives us a very nice because I can also take v transpose u V transpose u so if I transpose v that becomes v 1 v 2 v 3 And I multiply it by you which is u 1 u 2 u 3 just as we did with matrix Multiplication remember we wrote it like this and like this so it's this column this row Which is just u 1 v 1 plus u 2 v 2 plus u 3 v 3 So that gives us exactly you know That gives us a way to do dot products of two vectors as you take the second one Take this transpose put it in front of the first one and do normal Matrix see this is a matrix see that is a matrix the transpose and that Naturally follows from that this scalar or inner product or dot product of two vectors So in short, that's the transpose of a matrix. There's really nothing to it I don't think you will ever struggle with the transpose of a matrix ever struggle with a symmetric matrix and and a nice way to see The dot product or inner product by just just using the transpose