[ELI5] What is a column space and a null space of a matrix A?

Question

I’m just getting started with linear algebra and I’m not sure what column spaces or null spaces actually signify. What is the geometrical interpretation of both?

In: 1

Answer 1

You can consider the columns of a matrix as a **set** of (column) vectors. The space they span is the column space.

For example the matrix

1 2
3 4
5 6

has two column vectors, (1 3 5) and (2 4 6), which span a two-dimensional subspace of **R**^3

The null space is just the set of vectors going into 0 when you apply the matrix on them. If A is the matrix and x the vector, the null space consists of all vectors solution of the equation

A.x = 0

It’s easy to show that they form a subspace (of the space where you take your x-es).