Span — definition

WHAT is the span?

WHY this definition? Because the two operations a vector space gives you are scaling ($c\,v$) and adding ($u+v$). A linear combination $c_1v_1+\cdots+c_kv_k$ is just these two operations applied repeatedly. The span is therefore the closure — everything you can build using only the tools you're allowed.

HOW do you compute / reason about a span?

To check "is vector $w$ in $\operatorname{span}\{v_1,\dots,v_k\}$?", you ask: $\text{Do there exist scalars } c_i \text{ such that } c_1v_1+\cdots+c_kv_k = w?$ This is just a system of linear equations in the unknowns $c_i$. If it has a solution → yes; if not → no.

Why span is a subspace (mini-derivation from scratch)

We claim $S = \operatorname{span}\{v_1,\dots,v_k\}$ is a subspace. We must show: (1) contains $\mathbf 0$, (2) closed under addition, (3) closed under scaling.

1. Zero: take all $c_i=0$, giving $\mathbf 0 \in S$. ✔ Why this step? The empty-everything combination is a valid combination.
2. Addition: take $x=\sum a_i v_i$ and $y=\sum b_i v_i$. Then $x+y = \sum (a_i+b_i)v_i,$ which is again a linear combination → in $S$. ✔ Why? We grouped like terms; $(a_i+b_i)$ is just another scalar.
3. Scaling: $\lambda x = \lambda\sum a_i v_i = \sum (\lambda a_i)v_i \in S$. ✔ Why? Pulling $\lambda$ inside still leaves a linear combination.

So every span is a subspace — this is the deep reason spans matter.

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

• Linear combination — the building block of every span.
• Linear independence — controls whether extra vectors enlarge the span.
• Subspace — every span is one; spans are the standard way to produce subspaces.
• Basis — a minimal spanning set (spans + independent).
• Dimension — the number of vectors in a basis = "size" of the span.
• Column space — the span of a matrix's columns.
• Determinant — tests full-span in square cases.

Concept Map

Hinglish (regional understanding)