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Question: Prove that if {$A_1,A_2,\cdots,A_k$} is a linearly independent subset of $M_{n\times n} (F)$, then {$A^t_1, A^t_2,\cdots, A^t_k$} is also linearly independent.

My attempt: I know that the rank of a matrix and its transpose are same, and that a matrix can be converted into its transpose by elementary row and column operations. But my problem is, the same operation will not convert each matrix into its transpose! So if I consider a linear combination $a_1A_1+a_2A_2+\cdots+a_kA_k=0$, how do I show that $c_1A^t_1+c_2A^t_2+\cdots+c_kA^t_k=0$, where the scalars in both cases are all zeroes?

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Use linearity of the transposition $(A+B)^t=A^t+B^t$ and $(A^t)^t=A$: if $\sum_{j=1}^ka_j A_j^t=0$ for some scalars $(a_j)_{j=1}^k$ then taking the transpose $\sum_{j=1}^ka_j (A_j^t)^t=0$. The LHS is $\sum_{j=1}^ka_j A_j$ and we conclude by the assumption on the family $(A_j)_{j=1}^k$.

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