Dimension

31. Dimension

The dimension of a vector space $𝑉$ is the number of vectors in any basis of $𝑉$ .

dim (𝑉) = (number of vectors in a basis)

This number doesn’t depend on which basis you pick — every basis of the same vector space has the same size.

Example

$dim (ℝ^{𝑛}) = 𝑛$ (the standard basis $𝑒_{1}, \dots, 𝑒_{𝑛}$ has $𝑛$ vectors)
$dim$ of the $𝑥 𝑦$ -plane in $ℝ^{3}$ is $2$
$dim ({𝟎}) = 0$ (the trivial space has the empty basis)
$dim$ of the space of $𝑚 \times 𝑛$ matrices is $𝑚 𝑛$

31.1. Properties

For subspaces $𝑈, 𝑊 \subseteq 𝑉$ :

$dim (𝑈) \leq dim (𝑉)$
$dim (𝑈) = dim (𝑉)$ implies $𝑈 = 𝑉$ (no proper subspace of full dimension)
$dim (𝑈 + 𝑊) = dim (𝑈) + dim (𝑊) - dim (𝑈 \cap 𝑊)$ (Grassmann formula)
For a linear transformation $𝑇 : 𝑉 \to 𝑊$ :
$dim (𝑉) = dim (ker (𝑇)) + dim (im (𝑇))$
This is the Rank–Nullity Theorem

31.2. Computing dimensions of common subspaces

For an $𝑚 \times 𝑛$ matrix $𝐴$ :

$dim (Col(A)) = rank (𝐴)$
$dim (Null(A)) = 𝑛 - rank (𝐴)$ (the nullity)
$dim (Row(A)) = rank (𝐴)$ (column rank = row rank)

31.3. See also

Basis — what dimension counts
Rank — dimension of column / row / image
Rank–Nullity Theorem