Null Space

We discussed column space of a matrix $A$ as a vector space of linear combination of all of it's column vectors.
Null space is completely different thing.

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For a matrix say $A$ the Null space is the space of all $\vec{x}$ that solves $A\vec{x}=\vec{0}$

Example in 4 Dimension

Consider a matrix $A=\begin{bmatrix} 1 & 1 & 2\\ 2 & 4 & 6\\ 3 & 6 & 9\\ 4 & 8 & 12\\ \end{bmatrix}$
Null space for $A$ is space of all the $\vec{x} = \begin{bmatrix} x_1\\x_2\\x_3\\ \end{bmatrix}$ that solves $A\vec{x} = \vec{0}$
$Ax= \begin{bmatrix} 1 & 1 & 2\\ 2 & 4 & 6\\ 3 & 6 & 9\\ 4 & 8 & 12\\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ \end{bmatrix}$
So you can see our column space is $\mathbb{R}^4$ and the space we desire(Null space) is a subspace of $\mathbb{R}^3$.

What is the obvious value of $\vec x$ that solves $A\vec x=\vec 0$?

$$\vec x=0$$

What are some other value of $\vec{x}$ that solves $A\vec{x}=0$?
$\vec{x}=\begin{bmatrix} -1 \\ -1 \\ 1 \\ \end{bmatrix}$ solves $A\vec{x}=0$,
and $\vec{x}=\begin{bmatrix} -2 \\ -2 \\ 2 \\ \end{bmatrix}$, $\vec{x}=\begin{bmatrix} -3 \\ -3 \\ 3 \\ \end{bmatrix}$, $\vec{x}=\begin{bmatrix} -4 \\ -4 \\ 4 \\ \end{bmatrix},...$ also solves $A\vec{x}=0$.
We can summarize it as,
$\vec{x}=a\begin{bmatrix} -1 \\ -1 \\ 1 \\ \end{bmatrix};\quad a\in\mathbb{R}$ solves $A\vec{x}=0$.
What is this form ($a\vec{v}$) it's a line, so Null space of $A$ is a line in $\mathbb{R}^3$.

How can we assure that the space we get by $A\vec{x}=0$ is a vector space?

We need to show:

• $1.$ Sum of two vectors in Null Space remains inside the null space.
• $2.$ If we multiply a vector of Null space the resultant vector remains in that subspace

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Say we have two vectors $\vec{v}$ and $\vec{w}$ in the Null space.
So $A\vec{v}=\vec{0}$ and $A\vec{w}=\vec{0}$
$\Rightarrow A\vec{v} + A\vec{w}=\vec{0}$
$\Rightarrow A(\vec{v} + \vec{w})=\vec{0}$
so $\vec{v}+\vec{w}$ is inside the Null space.

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Say we have a vectors $\vec{v}$ in the Null space.
So $A\vec{v}=\vec{0}$
Say $\vec{w}=c\vec{v};\quad c\in\mathbb{R}$
$A\vec{w}=A(c\vec{v})=cA\vec{v}=c\vec{0}=\vec{0};\quad\{\text{because }A\vec{v}=\vec{0}$
So $\vec{w}$ is also in Null space.

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This confirms that Null Space is a vector space.