linear algebra - linear transformation and transpose

Every linear transformation can be represented as a matrix. If I take a matrix representation of a certain linear transformation and transpose it, what kind of operator do I get? when does the transpose operation preserves the mapping?

P.S. we are considering a square matrix $A\in \mathbb R ^{n \times n}$ which represents the mapping

$$L:\mathbb R^n \to \mathbb R^n$$

Solution 1:

If you are speaking in $\mathbb R^n$ the transpose gives you the adjoint. The adjoint satisfies the inner product inequality given by $$\left\langle {Ax,y} \right\rangle = \left\langle {x,{A^*}y} \right\rangle $$ Obviously $A^*=A^T$, where $A\in \mathbb R^{n \times n}$ is your matrix operator. Does it change or not depends on the linear transformation and hence A itself. if $A=A^T$ it does not change (like in the case of the identity operator). Hope this answers your question or part of it.