Transpose of a matrix

Let's say that I have the matrix A. And we know we denote matrices with bolded capital letters. So I have the matrix A. And it is a 3 by 2 matrix. And so let's say it's 1, negative 2, 3, 0, 7, 5. It's a 3 by 2 matrix. And what my question to myself-- or to you, if you're listening-- is what is the transpose of A? And so we denote the transpose like this, A with this little superscript T right over here. And all this is talking about is the transpose of matrix A. And so your natural question should be, well, what does it mean to take the transpose of a matrix? It sounds very fancy. Well, it is not that fancy. All this means is rows become columns. And you could imagine, when the rows become columns, the columns become the rows. What do I mean by that? Well, let's do it right over here. So if I have three rows. So let's be clear. This is one, two, three rows. So this is three rows by one, two columns. So this is a 3 by 2 matrix. If all the rows are going to become columns and all the columns are going to become rows, the transpose is going to be a 2 by 3 matrix. It's going to be two rows. The two columns here are going to become the two rows, by three columns. These three rows are going to become three columns. And so it's going to look something like-- let me do it in that blue color. It's going to have two rows now. And it's going to have three columns. So I said all the rows became columns. All the columns became rows. Well, I could view it exactly like that. I have this first row here of 1, negative 2. I can turn that into my first column now of 1, negative 2. I have the second row, 3 and 0. That will be now my second column, 3 and 0. And I have this third row, 7 and 5. Well, that's going to be my third column now, 7 and 5. And you could have just as easily viewed it the other way around. You could say, look, I have two columns. My first column is 1, 3, 7. So my first row now is going to be 1, 3, 7. And then you could say, my second column is negative 2, 0, 5. And so my second row will be negative 2, 0, 5. Let's do another example of that, just so we really make it clear. So let's say I have matrix B. Once again, bolded capital letter. And let's just say it's a really simple 2 by 2 matrix. So it's negative 1, 5, pi, and 3 are its entries. What is B transpose going to be equal to? Well, once again, we have two rows and two columns. So if all the rows become columns and all the columns become rows, I'm still going to have a 2 by 2 matrix. But we just have to say, look, my first row here is negative 1, 5. So that's now going to be my first column, negative 1 and 5. I have my second row here, which is pi and 3. So now that's going to become my second column of pi and 3. And so I have taken the transpose. All the rows became columns. All the columns became rows.