Let be an finite-dimensional vector space over a field . A **basis** for is a set such that

- is linearly independent, and
- .

For example, take , that is, the set of -tuples of elements of . Then the standard basis for is where , , , . It is clear that this set is linearly independent, as

implies that for each , and hence . If , then

A nonempty set is said to be a **subspace** of if is itself a vector space. Since any subset of inherits the operations of addition and scalar multiplication from , the associative and distributive laws follow automatically. Therefore is a subspace of if and only if

- is closed under addition, i.e. implies , and
- is closed under scalar multiplication, i.e. , implies .

Note that the second condition implies that , as for any , (this is a necessary condition for to be a subgroup of ). For an example of a subspace, let , and

For if then

and if then

One last definition: if , are vector spaces over a field , a function is called a **linear operator** if

- For all , , and
- For all and , .

If in addition is a bijection, then is called an **isomorphism** of vector spaces. The next post on linear algebra will show that all finite-dimensional vector spaces of the same dimension are isomorphic.