We define bases and consider examples of bases of and subspaces of .

VSP-0030: Introduction to Bases

Coordinate Vectors

When we first introduced vectors we learned to represent them using component notation. If we know that the head of is located at the point . But there is another way to look at the component form. Observe that can be expressed as a linear combination of the standard unit vectors and : In fact, any vector of can be written as a linear combination of and : This gives us an alternative way of interpreting the component notation:

We say that and are coordinates of with respect to , and is said to be the coordinate vector for with respect to . Every vector of can be thus represented using and . Moreover, such representation in terms of and is unique for each vector, meaning that we will never have two different coordinate vectors representing the same vector. We will refer to as a basis of .

The order in which the basis elements are written matters. For example, is represented by the coordinate vector with respect to , but changing the basis to would change the coordinate vector to .

Clearly, standard unit vectors and are very convenient, but other vectors can also be used in place of and to represent .

The diagram below shows together with vectors and .

Using Procedure pro:lincombgeo of VEC-0040, we can determine that as shown below.

If, for the sake of argument, we declare to be a basis of , then we can say that the coordinate vector for with respect to is .

What Constitutes a Basis?

In the previous section we had used the term basis without defining it. Now is the time to pause and think about what we want a basis to do. Let’s focus on and subspaces of . What we establish here will easily generalize to other vector spaces.

Based on our previous discussion, given any vector of (or a subspace of ), we want to be able to write a coordinate vector for with respect to the given basis of (or ). Based on this condition, we will require that basis vectors span (or ).

For example, consider and shown below.

The set cannot be a basis for because and span a plane in , and any vector not in that plane cannot be written as a linear combination of and .

On the other hand, the plane spanned by and is a subspace of . Because every vector in that plane can be written as a linear combination of and , the set can potentially be a basis for the plane, provided that the set satisfies our second requirement.

Our second requirement is that for a fixed basis of (or ), the coordinate vector for each in (or ) should be unique. Uniqueness of representation in terms of the basis elements will play an important role in our future study of functions that map vector spaces to vector spaces.

The following theorem shows that the uniqueness requirement is equivalent to the requirement that the basis vectors be linearly independent.

Suppose that every in can be expressed as a unique linear combination of . This means that has a unique representation as a linear combination of . But is a representation of in terms of . Since we are assuming that such a representation is unique, we conclude that there is no other. This means that the vectors are linearly independent.

Conversely, suppose that vectors are linearly independent. An arbitrary element of can be expressed as a linear combination of : Suppose this representation is not unique. Then there may be another linear combination that is also equal to : But then This gives us Because we assumed that are linearly independent, we must have so that This proves the representation of in terms of is unique.

Definition of a Basis

Bases are not unique. For example, we know that vectors and form the standard basis of . But, as we discussed in Example ex:spanr2 of VEC-0090, vectors are linearly independent vectors that span . Therefore is also a basis for .

Any linearly independent spanning set in (or a subspace of ) is a basis of (or the subspace). The plural form of the word basis is bases. It is easy to see that and its subspaces each has infinitely many bases.

Practice Problems

Let be a basis for . (Do a mental verification that is a basis.) For each given below, find the coordinate vector for with respect to .
Vector .


Vector .


Let be a basis for . Find the coordinate vector for with respect to .


Suppose is a basis for . Find if the coordinate vector for is .


Which of the following is a basis for ?

Which of the following is a basis for ?