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# Transpose of a matrix

## Video transcript

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.