Null Space

It is the set of vectors that satisfy $A\vec{x}=\vec{0},A\in M_{m\times n}(\mathbb{R})$

• So any vector that, after transformation through $A$, lands on $\vec{0}$ is part of this set.
• By definition, $\text{Null}(A)$ is a subset of $R^n$.

$\text{Null}(A)$ is a subspace of $R^n$ cuz:

• $\vec{0}$ is part of this set.
  • Cuz $\vec{0}$ satisfies this $A\vec{x}=\vec{0}$ equation.
  • $\vec{0}$ satisfies the equation, cuz the origin remains unchanged in a linear transformation (represented by the matrix).
• $\text{Null}(A)$ is closed under linear combination.
  • We have two vectors, who land on $\vec{0}$ after transformation.
  • The resultant of them will also land on $\vec{0}$ after transformation.

Theorem 28.4

See reference [2] for complete proof.

• It is a subspace of $R^n$, with $n$ specifically instead of $m$ because for matrix-vector product to work, the number of entries of the vector should be the same as the number of columns of the matrix.
• Since the matrix is $n$ columns wide, the vector is also $n$ columns tall, hence belonging in $R^n$.