Prove that the $n^{\text{th}}$ column of a matrix is pivotal if and only if it is linearly independent of all previous columns $i, i < n.$
Note (definition): a column of a matrix is pivotal if the index is pivotal in the reduced row echelon form of the matrix.
To prove the $\implies$ direction, since we know the column can be pivotal only if its index is pivotal which means it is the first column with a nonzero in a new row in the reduced row form, the column has to be independent from previous columns. To prove the other direction, if column $n$ is independent with previous columns, that means column $n$ has to be the new pivotal column according to the above note/definition.
Is this proof missing something? If not, is there a more elegant way of expressing this proof? (I feel like mine is wordy and not concise)