Subspace

A Vector Space’s version of a Subset, with some constraints.

While, a subspace does inherit the axioms of a Vector Space. Verifying a subset is a subspace is simpler; A Subset $H$ of ${\mathbb{R}}^n$ is a subspace if it is closed within scalar multiplication and vector addition:

$$\begin{array}{c}\text{For}c\in \mathbb{R},\vec{u},\vec{v}\in H\\ H\text{ is a subspace}\\ ⟺\\ c\vec{u}\in H\wedge \vec{u}+\vec{v}\in H\wedge \vec{0}\in H\end{array}$$

Properties

• $\text{span}H=H$, i.e. the Span of any subset is identical to itself

Four Fundamental Subspaces

For Matrix $A\in {\mathbb{R}}^{m\times n},r=\text{rank}A$

• The Subspaces are linked by the Fundamental Theorem of Linear Algebra
  • Rowspace
  • Nullspace of $A$
  • Columnspace
  • Nullspace of $A^T$, “Left Nullspace”
• Shows the relationship between the dimensions of the subspaces
  • $\dim \text{Col }A=r$
  • $\dim \text{Row }A=r$
  • $\dim \text{Nul }A=n-r$
  • $\dim \text{Nul }{A}^T=m-r$
• Shows the Morphisms between different subspaces of a matrix
  • Show existence of solutions:
    • $T(x):{\mathbb{R}}^n\to {\mathbb{R}}^m\equiv Ax$
    • $\exists x\in {\mathbb{R}}^n,Ax=b⟺b\in \text{Col }A$
• Shows the Orthogonality relationship between related subspaces of a matrix
  • $(\text{Row}A{)}^{\perp }=\text{Nul}A$
  • $(\text{Col}A{)}^{\perp }=\text{Nul}(A^T)$