Linear Algebra 63, Properties of Matrix Transpose, Proofs

[maxpercer]

I think in the proof for 1, it may help to define a_ij + b_ij = c_ij.
Then (A+B) ^T = (a_ij + b_ij) ^T = (c_ij)^T = c_ji = a_ji + b_ji = (a_ij)^T + (b_ij)^T = A^T + B^T.

[zaccandels]

In trying to prove that the operation of transposition of matrices is distributive you use the property itself. You simply state that (A+B)^T = a_{ji) + b_{ji} and don't tell us why this has to be the case

[abdullaalmosalami]

I feel like the first proof, you used the theorem to prove itself, which is circular and is not correct.