2.1311 min read

Basis: Minimal Spanning Sets

A basis for a subspace $V$ is a set of vectors that (1) spans $V$ and (2) is linearly independent. It's the "just right" description of $V$ — enough to generate everything, but no redundancy.

The canonical example is the standard basis of $\mathbb{R}^m$ : the vectors $\mathbf{e}_1 = (1,0,\ldots,0)$ , $\mathbf{e}_2 = (0,1,\ldots,0)$ , etc. Every vector in $\mathbb{R}^m$ is uniquely expressible as a linear combination of the $\mathbf{e}_i$ .

Any vector in $V$ is expressible as a unique linear combination of the basis vectors. This uniqueness is what makes a basis so powerful: it gives a coordinate system for $V$ .

Formal View

Definition 2.13 — Basis

A set

\{\mathbf{b}_1, \ldots, \mathbf{b}_k\}

is a basis for a subspace

V

if: \begin{enumerate} \item It spans

V

V = \text{Span}(\{\mathbf{b}_1, \ldots, \mathbf{b}_k\})

\item It is linearly independent. \end{enumerate} Equivalently, every

\mathbf{v} \in V

has a unique representation

\mathbf{v} = \sum_i c_i \mathbf{b}_i

Why This Matters

A basis is the minimal encoding of a subspace — it's the "DNA" of the space.

Fourier basis: any periodic signal is a unique combination of sine and cosine waves
Principal components in PCA are an orthogonal basis for data variation
Color mixing: RGB is a basis for displayable colors

Learning Resources

Basis and dimension

3Blue1Brown

Geometric intuition for basis vectors and coordinate systems.

10 min

Quiz

Question 1

Which of the following is a basis for $\mathbb{R}^2$ ?

Common Mistakes

Including too many vectors in a basis — a basis for $\mathbb{R}^n$ has exactly $n$ vectors, never more or fewer.
Thinking the standard basis $\{\mathbf{e}_i\}$ is the only basis — there are infinitely many bases for any subspace.