Dimension of a vector space

The dimension of a vector space is equal to the number of elements in a basis of that vector space. If the basis is infinite, we say that the dimension is infinite too.

Definition of the dimension of a vector space

Let V be a vector space and B a basis of it. That is, B ⊆ V, the set B contains linearly independent vectors and the span of B is equal to V: span(B) = V.

If the basis B has a finite number n of elements, we say that the dimension of V is n. We write the dimension as dim, so dim V = |B| = n. If the basis B is infinite, we say that the dimension of the space is also infinite. Since all bases of a space always have the same size, it does not matter which basis we take.

Example: the space ℝ³ has the basis (1, 0, 0), (0, 1, 0), (0, 0, 1). That is three vectors in total, so dim ℝ³ = 3. In general, dim ℝⁿ = n.

If a space V has dimension n, we speak of an n-dimensional vector space.

Properties of the dimension of a vector space

• Let V be a vector space and W ⊆ V a subspace of it. Then dim W ≤ dim V.
• Let V be a vector space of finite dimension and W ⊊ V a subspace of it. Then dim W < dim V. This point differs in that we do not allow W = V. So if a subspace W is smaller than V, it also has a lower dimension. Why? Because there is a vector x ∈ V ∖ W that is not a linear combination of vectors from W, and so it is not a linear combination of vectors from any basis of W either.
• If a space V has dimension n, then any n linearly independent vectors form a basis of V. A set of n + 1 vectors is then linearly dependent.