[Linear Algebra] Subspace, Basis

Span and Subspace

• Definition : A subspace $H$ is defined as a subset of $\mathbb{R}^{n}$ closed under linear combination

  For any two vectors $\textbf{u}_1, \textbf{u}_2$ and any two scalars $c, d$, $c\textbf{u}_1 + d\textbf{u}_2 \in H$

• $\textrm{Span}\begin{Bmatrix} \textbf{v}_1, \cdots, \textbf{v}_p \end{Bmatrix}$ is always a subspace
• Subspace is always represented as $\textrm{Span}\begin{Bmatrix} \textbf{v}_1, \cdots, \textbf{v}_p \end{Bmatrix}$

Basis of a Subspace

• Definition : a basis of a subspace $H$ is a set of vectors that satisfies both the following
  • Fully spans the given subspace $H$
  • Linearly independent (i.e. no redundancy)

Non-Uniqueness of Basis

• Consider a subspace $H$, green plane
• Basis of a subspace $H$ is not unique