We use the null space for the proof, therefore, I will explain the null space a bit. If you know about the null space, of course you can skip this entry.

The null space of a matrix

*A*is a set of non-zero vector

**x**that satisfies

*A*.

**x**=**0**Let me show you an example square matrix

*A*that has null space.

When

*, following*

**x**\neq**0***is a solution.*

**x**Therefore, this

*is a null space of*

**x***A*. When an

*a*is an scalar,

*a*are also the solutions. It means these are also null space. In this example, the matrix is singular (and square). If the matrix is not singular, the solution must be

**x**, a \neq 0*only. Because, if a square matrix is not singular, there is the inverse,*

**0**Therefore,

*. In this case, we say there is no null space.*

**x**=**0**Let me show you another example, but this time a rectangle matrix

*A*that has null space.

The solution is the same as the last example.

null space is related with a matrix is singular or not, that means the matrix has independent columns or not. Because, if we write down an

*A*with column vector, and multiply with

*,*

**x***A*

**x**is a linear combination of the column vectors

*with coefficients*

**a**_i*x_i*. If this result is

*,*

**0***is a null space. This is one aspect of null space.*

**x**When we look at the columns of

*A*,

**x**=**b***is the linear combination of column vector of*

**b***A*. A space consists of column vectors is called column space. If we have

*n*independent columns in

*n*-dimensional space, we could express any point in the

*n*-dimensional space. This means we have a solution for any

*and we have the inverse. If*

**b***A*has the inverse, the linear combination of the column vector becomes

*iff*

**0***. This also means there is no null space (= or only*

**x**=**0***exists).*

**0**Now we see the null space, column space, and the existence of the inverse. Do you think you have now a bit clearer view about the relationship among them?

Acknowledgements

Thanks to Marc D. who listened my explanation and gave me comments.