Span and Linear Combinations

Building with Vectors

A linear combination of vectors is an expression built by scaling and adding: $c_1 v_1 + c_2 v_2 + \dots + c_k v_k$, where $c_i$ are scalars. It is the fundamental way to construct new vectors.

The span $\operatorname{span}\{v_1, \dots, v_k\}$ is the collection of all possible linear combinations of those vectors. The span is always a subspace because it contains $\mathbf{0}$ and is closed under vector addition and scalar multiplication.

Spanning and Systems

A spanning set for a space $V$ is a set of vectors whose span is all of $V$. For example, $(1, 0)$ and $(0, 1)$ span $\mathbb{R}^2$ since any vector $(a, b) = a(1,0) + b(0,1)$.

To determine if $w$ is in the span of other vectors, solve the linear system $w = c_1 v_1 + \dots + c_k v_k$ for unknowns $c_i$. A solution exists if and only if $w$ lies in the span.

Common pitfall: Assuming more vectors always span more. Adding a vector that is already a linear combination of the existing ones does not enlarge the span. Only a vector pointing in a genuinely new direction expands the reach.