Transpose of a matrix

I've got a matrix A, and it's an m by n matrix. It has m rows and n columns. So I can write it in fairly general terms like this. The first row would be a11. First row, first column. a12, First row, second column. All the way to-- I have n columns. So a1n, first row, n-th column. And then the second row would look like this. a second row, first column. A second row, second column. All the way to a second row, n column. And we'll just keep doing that all the way down until you get to the m-th row. The m row would look like this. Each of these are the entries in each of the rows or columns, depending on how you want to look at it. So this is going to be a sub m 1. mth row, first column. a sub m 2. And you go all the way to a sub m n. This is our matrix right here. That is my matrix A. Now, I'm going to define the transpose of this matrix as a with this superscript t. And this is going to be my definition, it is essentially the matrix A with all the rows and the columns swapped. So my matrix A transpose is going to be a n by m matrix. Notice I said m rows and n columns. Now this is going to have n rows and m columns. So what is this guy going to look like? What is he going to look like? Well I'm gonna swap my rows and my columns. So my first row becomes my first column. So I'm going to have a11. That entry's still going to be in that position. But now this entry is not going to be right here. a12. And my second row I have what I used to have in my second row, first column. I'm now going to have what I had in my second column, first row. I'm just going to go down all the way to a1n. And that makes-- not a i n, a1n. And that makes sense because I'll now have n columns. Sorry I now have n rows. I had n columns before. Now I have n rows. Now this row, when I transpose it is going look like this. a21, a22, all the way down to a2n. It might be a little confusing for you right now to have this notation right there because everything we've done so far. We've always said, hey this first number's the row and the second number is the column. That's what we did up here. What I'm doing here, you can ignore that reference to the rows and columns. You can just say whatever we had here in my first row, second column, I now have here. When you look at this transpose, don't take these subscripts too literally. Or now you can kind of reverse your interpretation. This is now the first column, second row. This was the second row, first column. I don't want you get too confused with these subscripts. Just keep in mind, we're taking all the rows and turning them into the columns to get the transpose. And then you just keep doing this. And then this m-th row will now become the m-th column. am1, am2, all the way down to a m n. So, this entry is now that entry. If you know, this entry is now that entry. That entry is now at that entry. I think you get the idea. This is what a transpose is. And sometimes when you do in the abstract, it can be a little confusing. And we'll especially appreciate that once we do some of the proofs involving the transpose. But actually taking the transpose of an actual matrix, with actual numbers, shouldn't be too difficult. So, let's start with the 2 by 2 case. I'll try to color code it as best as I can. So let's say I have the matrix. Let's say I defined A. Let's do B now. I already defined A. Let's say B. B is equal to the matrix 1, 2, 3, 4. Those colors are pretty close. But what is B transpose going to look like? B transpose is going to be equal to-- You switch the rows and columns. So the first row will now become the first column. 1, 2. And the second row will now become the second column, 3, 4. Or you could view it the other way. The first column now became the first row. And the second column now became the second row. Let's do an example. Instead of even doing a 2 by 3-- or a 3 by 3-- let me do one that might be a little bit more challenging. I think this'll make things clear. So let's say I have the matrix C. Let me make it a pretty big matrix. Let's say it is a 4 by 3 matrix right here. Let me just throw some numbers in there. 1, 0, minus 1. 2, 7. Oh I want to do it in different colors. Let me do that in a different color. So then I get 2, 7, minus 5. Then I get 4 minus 3, 2. I have to do one more row here. So let me just make that minus 1, 3, and 0. That is my matrix C. So what is-- let me do that, and I like to be aesthetically pleasing. So let me close the bracket in the same color. So what is C transpose going to be? So, C transpose. Let me do that in a different color. C transpose is now going to be a 3 by 4 matrix. And, essentially, it's going to be the matrix C with all the rows swapped for the columns or all the columns swapped for the rows. So, it's now going to be a 3 by 4 matrix. And that first row there is now going to become the first column. 1, 0, minus 1. The second row here is now going to become the second column. 2, 7, minus 5. I didn't use the exact same green, but you get the idea. This third row will become the third column. 4, minus 3, 2. And then, finally, the fourth row will become the fourth column. Minus 1, 3, and 0. All we did is, this guy was in the second row, third column. Now, that same guy is in the what? He is in the second column and the third row. All we did is switch the rows and the columns. We could do it with another. Let's see. Let's do it with this one right here. This guy right here is in the third row. 1, 2, 3. And the second column. And when you go down here this guy is now in the third column and the second row. That's all a transpose is. And, just as a little interesting thing, what happens if we take the transpose of the transpose? So what happens if we take C transpose and then transpose that? What is that going to be equal to? Well to go from C to C transpose, we switched all the rows and the columns. All the entries the rows and columns. When you take the transpose again, remember let's just focus on this guy. This was second row, third column. You took the transpose, it becomes second column and third row. If you were to take the transpose again of that, he would then become the second row and third column again. So C transpose, the transpose of C transpose, is just equal to C. You're swapping all the columns when you take the transpose. And when you take the transpose again, you swap them all back. That's all that means. Anyway. Hope you found that useful.