Dimension

Created over 8 years ago, updated 10 days ago

Definition: The dimension of a vector space (V), denoted (\dim(V)), is the number of vectors in any basis of (V).

If (V) has a finite basis, it is called finite-dimensional. Otherwise, it is infinite-dimensional.

Examples:

(\dim(\mathbb{R}^n) = n) (standard basis has (n) vectors)
(\dim(\mathbb{C}^n) = n) over (\mathbb{C})
(\dim(P_n) = n+1) where (P_n) is the space of polynomials of degree at most (n) (basis: (1, x, x^2, \ldots, x^n))
(\dim(M_{m \times n}) = mn) for (m \times n) matrices
(\dim({\mathbf{0}}) = 0)

Key theorems:

If (\dim(V) = n), any set of (n) linearly independent vectors is a basis
If (\dim(V) = n), any spanning set of (n) vectors is a basis
If (W) is a subspace of (V), then (\dim(W) \leq \dim(V))
Rank-nullity theorem: For a linear transformation (T: V \to W), (\dim(V) = \dim(\ker T) + \dim(\text{im } T))