Question

Suppose `F` is a `5xx5` matrix whose column space is not equal to `RR^5` (`5` dimensions) . What can be said about null `F`?

Answer 1

The dimension of `"null"(F)` is `5-"rank"(F) > 0`

Explanation:

A `5xx5` matrix `F` will map `RR^5` to a linear subspace, isomorphic to `RR^n` for some `n in {0, 1, 2, 3, 4, 5}`.

Since we are told that this subspace is not the whole of `RR^5`, it is isomorphic to `RR^n` for some integer `n` in the range `0`-`4`, where `n` is the rank of `F`. Such a subspace is a `4` dimensional hyperplane, `3` dimensional hyperplane, `2` dimensional plane, `1` dimensional line, or `0` dimensional point.

You can choose `n` of the column vectors which span this subspace. It is then possible to construct `5-n` new column vectors which together with the `n` original ones span the whole of `RR^5`.

Then the `5-n` new column vectors span the null space of `F`.

In other words, the dimension of the null space of `F` is `5-"rank"(F)`.