Row rank, column rank, and scalar extension

Let \(M\) be a matrix over a field \(F\).

1. Prove that the row rank of \(M\) equals its column rank.
2. Over \(F=\mathbb R\), find a maximal linearly independent set of columns of \[ \begin{pmatrix} 1&0&3&-2\\ 2&1&2&0\\ 0&1&-4&4\\ 1&1&1&2\\ 1&0&1&2 \end{pmatrix}. \]
3. If \(F\subset K\) is a field extension and \(M\) has entries in \(F\), compare the row rank of \(M\) over \(F\) and over \(K\).