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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 a1 and first row and 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 and column and we'll just keep doing that all the way down until you get to the m p-- row the M throw 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 M throw first column a sub M 2 and you go all the way to a sub M and this is our matrix right here this is that is my matrix a now I'm going to define the transpose of this matrix I'm going to define the transpose of this matrix transpose transpose of this matrix as a with this superscript T and it's essentially and this is going to be my definition it's 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 notice to set 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 where I'm going to swap my rows in my columns so my first row becomes my first column so I'm going to have a 1 1 that entries still going to be in that position but now this entry is not going to be right here a 1 2 in my second row I have what I used to have in my sec in my second row first column I'm now going to have I had in my second column first row I'm just going to go down all the way to a 1n and that makes this not a I n a 1 and that makes sense because I'm going to have n columns I had n I sorry I now have n rows I had n columns before now I have n rows now this row when I transpose it when I transpose it's going to look like this a 2 1 a 2 2 all the way down to a 2n and if you could 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 is 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 so 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 so these are the I don't want you get too confused with these subscripts we're just keep in mind we're taking all of the rows and turning them into the columns to get the transpose and then you just keep doing this you just keep doing this and then this M row will now become the empty column am-1 am-2 all the way down to a m n all the way down to a MN so this entry is now that entry if you know this entry is now that entry that entry is now this entry I think you get the idea this is what a transpose is and sometimes when you do it 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 and I'll try to color code it as best as I can so let's say I have the matrix let's say I define a let's do B now find a let's say B B is equal to the matrix let's say it's equal to the matrix one two three four those yellows are pretty close but what does B transpose going to look like B transpose is going to be equal to you switch the rows in the column so the first row will now become the first column one two and the second row will now become the second column three or 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 let's head of even doing a two by three let me or three by three let me do one that might be a little bit more a little more challenging I think this will make things clearer so let's say I have the matrix C let me say I have the matrix C right here and let me make it a pretty big matrix let's say it is a four by three matrix right here so let me just throw some numbers in there one zero minus one to seven I want to do it in different colors let me do that in a different color so then I get to seven minus five then I get four minus three two and I have to do one more row here so let me just make that minus 1 3 and zero that is my matrix C so what is let me do that in I like to be aesthetically pleasing so let me close the bracket in the same color so what does 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 so now I'm going to be a 3 by 4 matrix and that first row there is not going to become the first column 1 0 minus 1 the second row here is now going to become the second column to seven minus five I didn't use the exact same green but you get the idea this third row will become the third column four minus three two and then finally the fourth row will become the fourth column minus one three and zero so all we did is if this guy was let's say this guy was in the position this guy was in the second row second row second row third column third column now that same guy is in the what he is in the second column second column and the third row the third row all we did is switch the rows and the columns we could do with another let's see let's do it with this one right here this guy right here is in the third row third row one two three and the second column second column and when you go down here this guy is now in the third column third column and the second row 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 with the rows and the columns but when you take the transpose again remember let's just focus on this guy this was second row third column you took the transpose he 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 is 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 hopefully you found that useful