Proving transpose(AB)=transpose(B)*tranpose(A) Help Needed

[torquerotates]

I need help on the tranpose of a multiple of a matrix.

I need to prove: transpose(AB)=transpose(B)*tranpose(A)

Any Ideas?

 

[HallsofIvy]

What is the definition of "transpose"?

 

[torquerotates]

Well, from what I know, I just switch the subscripts and interchange rows and columns. So if A=a(ij) then, transpose(A)=a(ji)

 

[torquerotates]

I just don't know where to go from there

 

[masnevets]

Did you try writing out the summation for the ij-th entry of both sides? I think it should be clear once you do that.

 

[HallsofIvy]

Recall that "transpose" is not only defined for square matrices. If A is an n by m matrix and B is an m by p matrix, so that you can multiply them, then A^T is an m by n matrix and B^T is a p by m matrix. If n is not equal to p, you can't multiply A^TB^T. But you can multiply B^TA^T.

 

[mathwonk]

if you know what it means in terms of being the induced map, namely composition, on dual spaces, the result is just the obvious fact that f*g*(h) = f*(g*h) = f*(hog) =
hogof = (gof)*(h) = (gf)*(h), so (gf)* = f*g*.

 

[torquerotates]

Well, I'm only taking an intro to linear algebra course. So I've never heard of an induced map.

 

[nishsweet]

How do you go about proving that the determinant of a nxn matrix A is equal to the determinant of the transpose of said matrix A using Laplace's expansion?

How can you use Det(AB) =Det A x det B to help with this?

 

[d_b]

what you need to do is to do a matrix multiplication for an abitrary matrix A and matrix B, I should say a transpose multiplication of matrix A and then transpose of matrix B. Then find the multiplication of matrix AB and find the transpose of that. It should be the same and that should do it.