[Linear_algebra] Null space

Let A be a linear transformation, where A ∈ R^(m×n).
The set of all vectors x such that A x = 0 is called the null space of A.

In geometric intuition, the null space represents the set of directions that get "squished" to the zero vector by A.

For example, let
A = [[1, 2], [3, 6]].

In this case, the null space is
{ [x, y]ᵗ ∈ R² | x = -2y },
or equivalently,
{ [-2y, y]ᵗ | y ∈ R }.

One example from this space is x = [-2, 1]ᵗ,
and you can verify that A x = 0.