3.4a Vector Spaces

Vector Spaces

A vector space $\mathcal{X}$ is a set of vectors that is closed under addition and scalar multiplication, such that it remains unchanged when those operations are performed. Formally, for any $u,v\in \mathcal{X}$ and $\alpha ,\beta \in \mathbb{R}$,
$\alpha u+\beta v\in \mathcal{X}.$

A nonempty subset $\mathcal{V}\subseteq \mathcal{X}$ of the vector space is called a subspace of $\mathcal{X}$ if it is also closed under addition and scalar multiplication.

An affine subspace is a subspace shifted away from the origin) $\mathcal{A}\subseteq \mathcal{V}$, where every elements satisfies
$\mathcal{A}=\{\alpha \vec{x}+(1-\alpha )\vec{y}:\vec{x},\vec{y}\in \mathcal{V},\alpha \in \mathbb{R}\}.$

Zero Vector always contain the zero vector (confirmed by setting $\alpha =0$ and $\beta =0$).

Both vector spaces and subspaces

Properties

(Sum of Subspaces) For all subspaces $\mathcal{U},\mathcal{V}$, their sum $\mathcal{U}+\mathcal{V}=\mathcal{W}$ is also a subspace.

(Direct Sum) For all subspaces $\mathcal{U},\mathcal{V}$, if $\mathcal{U}\cap \mathcal{V}=\{0\}$, then $\mathcal{U}\oplus \mathcal{V}=\mathcal{U}+\mathcal{V}=\mathcal{W}$ and every vector in $\mathcal{W}$ can be uniquely represented as $w=u+v$.

(Dimension of Direct Sum) For any $\mathcal{U},\mathcal{V}$, $\dim (\mathcal{U}+\mathcal{V})=\dim \mathcal{U}+\dim \mathcal{V}$.