Vector Spaces and Subspaces

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

Subspace test — $W \subseteq V$ is a subspace if:

1. $\vec{0} \in W$
2. Closed under addition
3. Closed under scalar multiplication

Key concepts

• Span: all linear combinations of a set of vectors
• Linear independence: no vector is a combination of the others
• Basis: a linearly independent spanning set
• Dimension: number of vectors in any basis

Fundamental subspaces of $A_{m\times n}$:

• Column space $\mathrm{Col}(A) \subseteq \mathbb{R}^m$
• 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 columns at pivot positions.

Common pitfall: A subspace must contain the zero vector — that is the fastest test to run. The solutions of $Ax = b$ with $b \ne 0$ never form a subspace, however plane-like they look.