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Dimension

The dimension of a subspace $V$ is the number of vectors in any basis for $V$ . This is well-defined because all bases for the same subspace have the same number of vectors.

Dimension captures how "big" a subspace is: $\dim(\{\mathbf{0}\}) = 0$ , $\dim$ (a line through origin) $= 1$ , $\dim$ (a plane through origin) $= 2$ , and $\dim(\mathbb{R}^m) = m$ .

The Steinitz Exchange Lemma guarantees that all bases have the same size: if $S$ spans $V$ and $T$ is linearly independent in $V$ , then $|T| \leq |S|$ . This is what makes dimension a well-defined invariant.

Formal View

Definition 2.14 — Dimension

The dimension of a subspace

V

, written

\dim(V)

, is the number of vectors in any basis for

V

Theorem 2.14 (Steinitz Exchange)

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

spans

V

and

\{\mathbf{a}_1, \ldots, \mathbf{a}_n\}

is linearly independent in

V

, then

n \leq k

. Consequently, all bases for

V

have the same cardinality.

Why This Matters

Dimension is the fundamental measure of a subspace's "size" and appears throughout applied mathematics.

Degrees of freedom in engineering systems = dimension of the solution space
Number of independent parameters in a statistical model = dimension of the model space
Rank of a matrix = dimension of its column space

Learning Resources

Dimension and rank

MIT OpenCourseWare

Gilbert Strang on dimension, rank, and the four fundamental subspaces.

49 min

Quiz

Question 1

A subspace $V \subseteq \mathbb{R}^5$ has dimension 3. How many vectors are in any basis for $V$ ?

Common Mistakes

Confusing the dimension of the ambient space ( $\mathbb{R}^m$ ) with the dimension of a subspace within it.
Thinking different bases can have different numbers of vectors — they cannot.