Vector Spaces and Subspaces

Vector Spaces & Subspaces

A vector space $V$ over $\mathbb{R}$ is a set equipped with addition and scalar multiplication satisfying eight strict algebraic axioms.

To prove a subset $W \subseteq V$ is a subspace, you only need three checks. Subspace test criteria must all hold:

Common pitfall: A subspace must contain the zero vector. The solutions of $Ax = b$ with $b \neq 0$ never form a subspace, however plane-like they look.

Core Concepts & Subspaces

Key concepts build the structural anatomy of vector spaces:

For any matrix $A_{m \times n}$, its fundamental subspaces are the Column space $\mathrm{Col}(A) \subseteq \mathbb{R}^m$ and Null space $\mathrm{Null}(A) \subseteq \mathbb{R}^n$.

Rank-nullity: $\mathrm{rk}(A) + \mathrm{null}(A) = n$. Tip: To find a basis for $\mathrm{Col}(A)$, row-reduce and take pivot columns.