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I am confused why the dimension of the column space of a matrix is the number of pivotal columns? Could anyone elaborate it by some concrete examples? BTW, could you also elaborate a bit about why number of pivotal columns = number of pivotal rows? Is it related to A and A^T? I don't quite understand what does pivotal mean here. Thanks a lot!

Jerry
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    This answer gives the explanation that I find clearest and easiest to understand: https://math.stackexchange.com/a/332945/169852 –  Oct 31 '20 at 08:47
  • Thank you so much! This is a very helpful answer! Thank you! – Jerry Oct 31 '20 at 08:56

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That the dimension of the column space equals that of the row space is a nontrivial theorem. The pivots are the leading $1$'s ($1$'s with only zeros below it), after row-reduction. Consider $\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$. After row-reduction/gaussian elimination, we get $\begin{pmatrix}1&2&3\\0&1&-2\end{pmatrix}$. There are two leading $1$'s, so two pivot columns, and the dimension of the column/row space is two. Thus we can say the rank is two.