1. The Two Demands Married: What Is a Basis?
Recall from the previous topic: We studied two properties of a set of vectors separately:
- Spanning: A set $S$ spans a vector space $V$ if every vector in $V$ can be written as a linear combination of vectors in $S$. (Enough vectors to reach everything.)
- Linear Independence: A set $S$ is linearly independent if no vector in $S$ is redundant — the only way to get the zero vector from a linear combination is to use all-zero coefficients. (No vector wasted.)
Definition — Basis
A basis for a vector space $V$ is a set of vectors $\mathcal{B} = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}$ that satisfies both demands simultaneously:
- Independence: $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n = \mathbf{0} \;\Longrightarrow\; c_1 = c_2 = \cdots = c_n = 0.$
- Spanning: For every $\mathbf{v} \in V$, there exist scalars $a_1, \ldots, a_n$ such that $\mathbf{v} = a_1\mathbf{v}_1 + \cdots + a_n\mathbf{v}_n.$
In short: a basis is a linearly independent spanning set — enough vectors to reach everything, no vector wasted.