How to visualize the rank of a matrix?

Question

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?

Answer 1

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...

Answer 2

Note that $m\times n$ matrices are in one-to-one correspondence with linear maps $\Bbb R^n\to\Bbb R^m$. This correspondence is given by $$ \begin{array}{ccc} \{m\times n\text{ matrices}\} & \leftrightarrow & \{\text{linear maps }\Bbb R^n\to\Bbb R^m\} \\ A &\mapsto& \left(\vec x\mapsto A\vec x\right) \\ [T]_\beta & \leftarrow & T \end{array} $$ Here $[T]_\beta$ is the matrix of $T$ relative to the standard basis $\beta$.

Now, the rank of an $m\times n$ matrix $A$ is the dimension of the image of the linear map $\vec x\mapsto A\vec x$. This gives the best geometric interpretation one could hope for.

For example, the linear map $\Bbb R^2\to\Bbb R^2$ given by $(x,y)\mapsto (x,0)$ is a linear map. One can visualize the image of this map as the projection onto the $x$-axis. The image is clearly one-dimensional so the rank is one.

Answer 3

Consider the following two equations in two variables, $$3x+5y=0$$ $$3x+5y=0$$

Now two equations in two variables should have only one solution, they should determine the solution uniquely. Ok, but one may object that the two equations that I have written down are the same, thus it is not really two equations but one equation written twice. The rank in other words is $1$. That is to say, these two equation really only impose one condition on the variables. The rank of a system of equations is the number of DISTINCT conditions which they impose of the variables. This can be smaller than the number of equations because one or more of the equations may be a combination of the others.

A matrix represents (among other things) the coefficients of a system of linear equations.