Define $T: M_{m \times n}(\mathbb{R}) \to M_{n \times m}(\mathbb{R})$ by $T(A)=B$ where $b_{ji}=a_{ij}$.
I am having a confusion and lost with the notation for proving that this is a linear transformation. I will indicate below.
Claim: $T$ is a linear transformation.
Proof:
Let $A, B$ be $m \times n$ matrices where $A = a_{ij} \in \mathbb{R}$ and $B = b_{ij} \in \mathbb{R}$.
$(1)$ First we want to prove that $T(A+B)=T(A)+T(B). \bigstar$
Version 1: Consider, $T((A+B))=(A+B)^T=(a_{ij}+b_{ij})^T=(a_{ij})^T+(b_{ij})^T=a_{ji}+b_{ji}=A^T+B^T=T(A)+T(B).$
Version 2: (Of the same proof above)
Consider, $T(A+B)=(A+B)^T=A^T+B^T=T(A)+T(B).$
In the proof above, I got from left hand side of the equation of $\bigstar$ to the right hand side of the equation. These two confuses me a little because how does one go from $(A+B)^T=A^T+B^T$? Why is this allowed? I feel that version 1 explains why $(A+B)^T=A^T+B^T$ but I'm not exactly sure how. I think this is an issue with the notation. Any clarfication would be appreciated, thank you in advance!
Version 2: definitely wrong. You have assumed that $(A+B)^T=A^T+B^T$, which is (part of) what you are supposed to prove.
Version 1: with $$(a_{ij}+b_{ij})^T=(a_{ij})^T+b_{ij})^T$$ you have done exactly the same thing. You could do better simply by leaving this out: $$T((A+B))=(A+B)^T=(a_{ij}+b_{ij})^T=a_{ji}+b_{ji}=A^T+B^T=T(A)+T(B)\ .$$ I still have a bit of a problem with this, as you appear to be saying that $A$, a matrix, is the same as $a_{ij}$, a single entry in that matrix. It doesn't hurt at all to use words in your proof, so how about something like this: $$\eqalign{ \hbox{$(i,j)$ element of $(A+B)^T$} &=\hbox{$(j,i)$ element of $A+B$}\qquad\hbox{(definition of transpose)}\cr &=\hbox{$(j,i)$ element of $A$}+\hbox{$(j,i)$ element of $B$}\cr &\qquad\qquad\qquad\qquad\hbox{(definition of matrix addition)}\cr \hbox{$(i,j)$ element of $A^T+B^T$} &=\hbox{$(i,j)$ element of $A^T$}+\hbox{$(i,j)$ element of $B^T$}\cr &=\hbox{$(j,i)$ element of $A$}+\hbox{$(j,i)$ element of $B$}\ .\cr}$$ Since this is true for all $i,j$, we have $(A+B)^T=A^T+B^T$.
BTW, as you are using $T$ to indicate transpose, it would be a really good idea to use a different letter for your linear transformation.