A transpose proof

  • #1

Main Question or Discussion Point

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

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

Any Ideas?
 

Answers and Replies

  • #2
HallsofIvy
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What is the definition of "transpose"?
 
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  • #3
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)
 
  • #4
I just don't know where to go from there
 
  • #5
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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.
 
  • #6
HallsofIvy
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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 AT is an m by n matrix and BT is a p by m matrix. If n is not equal to p, you can't multiply ATBT. But you can multiply BTAT.
 
  • #7
mathwonk
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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*.
 
  • #8
Well, I'm only taking an intro to linear algebra course. So I've never heard of an induced map.
 
  • #9
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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?
 
  • #10
d_b
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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.
 

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