How to visualize the rank of a matrix? Intuitively, what is the rank of a matrix? Slope is the steepness of a line. What does the rank of a matrix stand for and is there any way to visualize it?

The rank of the matrix gives the number of linearly independent column vectors of the matrix and this number also means the dimension of the linear space these vectors span.
For example:
$\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}$
This matrix has 3 column vectors which span the 3 dimensional space, they are it's trivial base vectors, so they are linearly independent...
