Transformation of A = transpose[A] for mxn Matrices

In summary, the task is to find the matrix representation of the transformation T, where T(A) = AT, for the standard bases of M2,3 and M3,2. The attempt at a solution involves determining the size of the transformation matrix B, which should result in a 3x3 matrix when multiplied by A. However, it is not possible to multiply a 3x2 matrix with a 2x3 matrix, so the solution may involve writing the matrices as single columns of 6 numbers instead.
  • #1
spybloom
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Homework Statement


"Let T:M2,3→M3,2 be represented by T(A) = AT. Find the matrix for T relative to the standard bases for M2,3 and M3,2"

Homework Equations


I let the transformation matrix be B. I know that BA = AT, so I need some matrix times A to equal A transpose.

The Attempt at a Solution


I'm having problems just trying to figure out the size of B. If B is 3x2, then the resultant matrix would be 3x3, which wouldn't equal the size of AT. I've tried playing around with adding a row of zeros to the bottom of A, but then I can't figure out the contents of B. Quite frankly, I'm stumped.
 
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  • #2
I presume that you know that if you multiply an m by n matrix by an n by p matrix, you get a m by p matrix. You cannot multiply any matrix by a 3 by 2 matrix and get a 2 by 3 matrix. In order to be able to write this transformation as a matrix, you will have to write your matrices as single columns of 6 numbers.
 

1. What is the definition of the transpose of a matrix?

The transpose of a matrix is the matrix that results from flipping the rows and columns of the original matrix. This means that the first row of the original matrix becomes the first column of the transposed matrix, the second row becomes the second column, and so on.

2. How is the transpose of a matrix calculated?

To find the transpose of a matrix, simply write out the original matrix and then switch the rows and columns. For an mxn matrix, the resulting transpose will be an nxm matrix.

3. What is the purpose of transposing a matrix?

The transpose of a matrix is often used to simplify calculations and equations involving matrices. It can also be used to convert a row vector into a column vector, or vice versa.

4. Can any matrix be transposed?

Yes, any matrix can be transposed regardless of its size or the values within it. However, the resulting transpose may not always be a valid matrix if the original matrix was not square.

5. How does transposing a matrix affect its properties?

The transpose of a matrix does not change its determinant, rank, or eigenvalues. However, it does change the trace and the sign of the eigenvalues. Additionally, the transpose of a matrix can affect the inverse of the matrix, as well as the null space and column space.

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