Define T Mnn R rightarrow Mnm R by TA AT where AT is the tr

Define T: M_nn (R) rightarrow M_nm (R) by T(A) = A^T, where A^T is the transpose of A. Is T a linear transformation? If it is, show it; otherwise, provide an example to disprove it.

Solution

Transposing of matrix is not a linear transformation as linear transformation is defined for vectors only not matrices.

It would have been linear transformation only if m=n that means T:M(n*n)-M(n*n) then only it can be linear transformation.

If m is not equal to n then no of columns become no of rows and no of rows become no of columns then this transformation can not be linear transformation.

Thank you