It is easy to see that as shown below.
If we declare to be a basis of , then we can say that the coordinate vector for with respect to is .
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 .
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 is a spanning set for a subspace of . Then every element of has a unique
representation as a linear combination of if and only if the vectors are linearly
independent.
- Proof
- 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.
Use , where to illustrate why a set of linearly dependent vectors cannot be used as a
basis for a subspace by showing that linearly dependent vectors fail to ensure
uniqueness of coordinate vectors for vectors in .
We will first show that the elements
of are linearly dependent. Let be a matrix whose columns are the vectors in
.
We find that
Therefore the matrix equation has infinitely many solutions:
This tells us that there are infinitely many nontrivial linear relations among the
elements of . Letting gives us one such nontrivial relation.
Now let’s pick an arbitrary vector in . Any vector will do, so let Based on this representation of , the coordinate vector for with respect to is But So, by substitution, we have: Based on this representation, the coordinate vector for with respect to is The set is linearly dependent. As a result, coordinate vectors for elements of are not unique and we do not want to use as a basis for .
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, 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.
Let . The set
is a basis for because the two vectors in are linearly independent and span . Find
the coordinate vector for with respect to .
We need to express as a linear
combination of the elements of . To this end, we need to solve the vector
equation:
The augmented matrix and the reduced row-echelon form are:
We conclude that , . This gives us
The coefficient in front of the first basis vector is , the coefficient in front of the second basis vector is . This means that the coordinate vector for with respect to is .
It may seem strange to you that the coordinate vector for a vector in only has two
components. But remember that subspace is a plane. When viewed as a vector in the
plane, it makes sense that the coordinate vector for only requires two components.
This issue is related to the question of dimension, which will be addressed in Bases
and Dimension.
To construct the coordinate vector for with respect to , we had to be mindful of the
order of the elements in . Ordinarily, the order of elements in a set is irrelevant, and
the basis
is considered to be the same as
When dealing with coordinate vectors, however, the order of the elements dictates
the order of the components of the coordinate vector coefficients. If we switch the
order of the elements in , the coordinate vector becomes . For this reason, when we
come back to studying coordinate vectors in more detail, we will use the term ordered
basis to avoid confusion.
