📐 Span is a Subspace

For any subset $S$ of a vector space $V$, $\\text{span}(S)$ is a subspace of $V$. Moreover, it is the smallest subspace containing $S$.

Proof: **Subspace:** Let $u = \\sum a_i v_i$ and $w = \\sum b_j u_j$ be in $\\text{span}(S)$.

- $u + w = \\sum a_i v_i + \\sum b_j u_j$ is a linear combination, so $u + w \\in \\text{span}(S)$.

- For $c \\in F$: $c \\cdot u = c \\sum a_i v_i = \\sum (ca_i) v_i \\in \\text{span}(S)$.

**Smallest:** Any s...

From: Advanced Linear Algebra

Learn more: https://advanced-linalg.vercel.app/#/section/3

Explore all courses: https://mathacademy-cyan.vercel.app