[LA] Null Space

Null Space는 주로 matrix 에 관련된 맥락에서 사용되며,

Linear Transformer의 맥락에서는 Kernel이라고 불림.

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Definition : Null Space

The null space of an $m \times n$ matrix $A$, written as Nul $A$, is the set of all solutions of the homogeneous equation $A\textbf{x}=\textbf{0}$.

In set notation,

$$\text{Nul }A = \left\{ \textbf{x}:\textbf{x} \text{ is in }\mathbb{R}^n \text{ and }
A\textbf{x}=\textbf{0} \right\}$$

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Theorem 2

The null space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^n$.
Equivalently, the set of all solutions to a system $A\textbf{x}=\textbf{0}$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $\mathbb{R}^n$.

• Proof
  • Nul $A$ is a subset of $\mathbb{R}^n$ because $A$ has $n$ columns. ($\textbf{x}$ is a $n \times 1$ matrix or $n$-entries vector)
  • We need to show that Nul $A$ satisfies the three properties of a subspace.
    • $\textbf{0}$ is in Nul $A$.
    • Next, let $\bf{u}$ and $\bf{v}$ represent any two vectors in Nul $A$.
      • Then
      • $$
        A\textbf{u}=\textbf{0} \text{ and } A\textbf{v}=\textbf{0}
        $$
      • To show that $\textbf{u} + \textbf{v}$ is in Nul $A$, we must show that $A(\textbf{u}+\textbf{v})=\textbf{0}$.
      • Using a property of matrix multiplication, compute
      • $$
        A(\textbf{u}+\textbf{v})= A\textbf{u} + A\textbf{v} = \textbf{0}+\textbf{0} = \textbf{0}
        $$
      • Thus $\textbf{u}+\textbf{v}$ is in Nul $A$, and Nul $A$ is closed under vector addition.
    • Finally, if $c$ is any scalar, thenwhich shows that $c\textbf{u}$ is in Nul $A$.
    • $$
      A(c\textbf{u}) = c(A\textbf{u}) = c(\textbf{0}) = \textbf{0}
      $$