# Fundamental Subspace of a Matrix

### Nullspace

Let $A∈M_{m×n}(R)$. The **nullspace** (also called kernel) of $A$ is the subset of $R_{n}$ defined by
$Null(A)={x∈R_{n}∣Ax=0}$

- the nullspace of A is simply the solution space of the homogeneous system of equations $Ax=0$ and is hence a subspace of $R_{n}$

"Find a basis for Null(A)"

Given the matrix $A$ in RREF form $ 100 010 320 010 $. The solution to the homogeneous system $Ax=0$ is $ x_{1}x_{2}x_{3}x_{4} =s −3−210 +t 0−101 $ So we have that $B_{1}={ uvw , uvw }$ is a basis for Null(A) and dim(Null(A)) = 2.

### Column Space

Let $A=[a_{1} … a_{n} ]∈M_{m×n}(R)$. The **column space** of $A$ is the subset of $R_{m}$ defined by
$Col(A)={Ax∣x∈R_{n}}=span{a_{1},…,a_{n}}$

Find a basis for Col(A)

To find a basis. to complete: page 167 of notes

### Row Space

Let $A= r_{1}⋮r_{m} ∈M_{m×n}(R)$

The **row space** of $A$ is the subset of $R_{n}$ defined by
$Row(A)={A_{T}x∣x∈R_{m}}=span{r_{1},…,r_{m}}$

Note System-Rank Theorem

$dim(Null(A))=n−rank(A)$