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.