# Null space

### Null space

#### Lessons

The null space of a matrix $A$ is the set $N(A)$ of all solutions of the homogeneous equation $Ax=0$

The null space of a matrix $A$ is a subspace of $\Bbb{R}^n$. The first question shows the proof of this.

To see if a vector $u$ is in $N(A)$ (nullspace of $A$), we simply compute:

$Au=0$

If the product of $A$ and $u$ gives the zero vector, then it is in the null space of $A$.

To find a basis for the null space of A, we have to:
1) Solve for $Ax=0$.
2) Write the general solution in parametric vector form.
3) The vectors you see is a basis for $N(A)$.

Note that the vectors in the basis are linearly independent.
• 1.
Null Space Overview:
a)
Definition of the Null space
$N(A) =$ null space
• A set of all vectors which satisfy the solution $Ax=0$

b)
A vector u in the null space
• Multiply matrix $A$ and vector $u$
$Au=0$ means $u$ is in the null space

c)
Finding a basis for the null space
• Solve for $Ax=0$
• Write the general solution in parametric vector form
• The vectors you see in parametric vector form are the basis of $N(A)$.

• 2.
Showing that the null space of $A$ is a subspace
Show that $N(A)$ (null space of $A$) is a subspace by verifying the 3 properties:
1) The zero vector is in $N(A)$
2) For each $u$ and $v$ in the set $N(A)$, the sum of $u+v$ is in $N(A)$ (closed under addition)
3) For each $u$ in the set $N(A)$, the vector $cu$ is in $N(A)$. (closed under scalar multiplication)

• 3.
Verifying a vector is in the null space
Is the vector in the null space of the matrix

• 4.
Is the vector in the null space of the matrix

• 5.
Finding a basis for the null space
Find a basis for the null space of $A$ if:

• 6.
Find a basis for the null space of $A$ if:

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5.
Subspace of $\Bbb{R}^n$
5.1
Properties of subspace
5.2
Column space
5.3
Null space
5.4
Dimension and rank