Learn Linear Algebra

Theorem

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 $A\vec{x} = \vec{0}$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $\mathbb{R}^n$.

Theorem

The column space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^m$.

Theorem

The column space of an $m \times n$ matrix $A$ is all of $\mathbb{R}^m$ if and only if the equation $A\vec{x} = \vec{b}$ has a solution for each $\vec{b} \in \mathbb{R}^m$.