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Linear Algebra: Companion notes

Abhay Soman

Section 3.3 Basis and Dimension

We are now in a position to give the definition of dimension of a vector space. We begin with the definition of a basis.

Definition 3.3.1. (Basis of a vector space).

Let \(V\) be a vector space over a field \(F\text{.}\) A basis of \(V\) over \(F\) is a set of linearly independent vectors in \(V\) which spans \(V\) (refer to Definition 2.5.13).
Now we define the dimension of a vector space.

Definition 3.3.2. (Dimension of a vector space).

Let \(V\) be a vector space over a field \(F\text{.}\) The cardinality of a basis of \(V\) is called the dimension of \(V\text{.}\) It is denoted by \(\dim_F V\) or by \([V:F]\text{.}\)
If \(\dim_F V\) is finite then, \(V\) is said to be a finite-dimensional vector space over \(F\) or if the context is clear then simply finite-dimensional.
If \(\dim_F V\) is infinite then, \(V\) is said to be an infinite dimensional vector space over \(F\) or if the context is clear then simply infinite dimensional.

Remark 3.3.3.

It may not be immediately clear from the definition of finite-dimensional vector space (Definition 3.3.2) that every basis has the same cardinality. This will be proven in Corollary 3.4.4.

Remark 3.3.4.

The dimension of a vector space \(V\) is an invariant of \(V\text{.}\) This will be made precise later.
A proof is based on an application of Zorn’s lemma.
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