Subspaces

Subspace

Definition 1(Definition).

A subspace of a vector space $V$ is a non-empty subset $V_0\subset V$ which is closed under the operations (addition, scaling) inherited from $V$.

The subspaces $V$ and $\{0\}$ are called trivial subspaces of $V$.

With Each linear transformation $A:V\to W$ we can associate the following two subspaces:

1. The null space, or kernel of $A$, which is denoted as $\text{Null }A$ or $\text{Ker }A$ and consists of all vectors $\mathbf{v}\in V$ such that $A\mathbf{v}=0$.
2. The range $\text{Ran }A$, the set of all vectors $\mathbf{w}\in W$ which can be represented as $\mathbf{w}=A\mathbf{v}$ for some $\mathbf{v}\in V$.