change of basis
where the sum is taken over a finite number of elements in . Suppose now that is another basis for . By a change of basis from to we mean re-expressing in terms of base elements .
Formally, we can think of a change of basis as the identity function (viewed as a linear operator) on a vector space , such that elements in the domain are expressed in terms of and elements in the range are expressed in terms of .
Note that, by the very design of a basis, a change of basis in a vector space is always possible.
where is the identity operator. is called a change of basis matrix. By applying to a vector expressed in terms of , we get expressed in terms of :
where and are expressed in the two bases and respectively.
Let and the following two sets
be the two ordered bases for , ordered in the way the elements are arranged in the set. For each , , we see that
Notice that the columns of are exactly the elements of . Indeed, each element of is already written in terms of the standard basis elements (in ). For example, let be the first basis element in . Let us see what is, when expressed using base elements in , the standard ordered basis:
exactly as we have expected.
Conversely, let be the first basis element in . What is when expressed in terms of basis elements of ? In other words, we need to find
Now, is just , so is nothing more than the first column of , which is just the inverse of the matrix , so
Therefore, . A quick verification shows that this is indeed the case:
Now let be the set . It is easy to check that forms a basis for (determinant is non-zero). Order in the obvious manner. What is the change of basis matrix ? One way is to express each element of in terms of the elements of . Another way is to use the formula . Applying the first example, we see that is just the matrix whose columns are elements of . As a result:
Remarks. Let us summarize what we have learned from the examples above, as well as list some additional facts. Let be a finite dimensional vector space of dimension .
If is the standard basis (ordered), then for any ordered basis , is the matrix whose columns are exactly the basis elements in (assuming these elements have already been expressed in terms of ) such that the -column corresponds to the -th element in the ordered set .
Continue to assume that is the standard basis. Let be any ordered bases for . Using the above property, we can easily compute , which is
Suppose is a linear transformation from to (both finite dimensional). Under a bases and , has matrix representation . Under changes of basis from to , and to , we have
If is a linear operator on , then setting , and from above, we have that
where is the change of basis matrix . This shows that and are similar matrices. In other words, under a change of basis, the linear transformation is basically the same.
- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
|Title||change of basis|
|Date of creation||2013-03-22 17:30:18|
|Last modified on||2013-03-22 17:30:18|
|Last modified by||CWoo (3771)|
|Synonym||change of coordinates|
|Synonym||change of bases|
|Synonym||base change matrix|
|Defines||change of basis matrix|