Given $T: M_{m\times n} \to M_{m\times n}$, defined by $T(A)=A^T$. Prove: $T$ is a linear transformation

Question

I'm having a lot of trouble with this problem. I just don't know where to start from. I know that for it to be a linear transformation it needs to satisfy two conditions $$T(u+v) = T(u) + T(v)$$ $$T(cu) = c\cdot T(u)$$

Any help would be great :)

Answer 1

Ok. So you already know what you need to check.

Is it true that $$T(A+B) = T(A) + T(B)?$$ Is it true that $$T(cA) = cT(A)?$$

You can prove these identities hold by showing that all entries of the matrix on the left hand side are the same as the corresponding entries of the matrix on the right. So for example, $$T(A+B)_{ij} = (A+B)_{ji}$$ by the definition of the transpose, and $$(A+B)_{ji} = A_{ji} + B_{ji}$$ since matrix addition is defined component-wise. On the other hand $$T(A)_{ij} = A_{ji}, \qquad T(B)_{ij} = B_{ji}$$ so $$T(A+B)_{ij} = A_{ji} + B_{ji} = (T(A)+T(B))_{ij}$$ for all values of $i$ and $j$, and therefore $$T(A+B) = T(A) + T(B).$$

Now you can apply a similar argument for the second identity.