change of basis
Let be a vector space. Given a basis for , each vector can be uniquely expressed in terms of the base elements as follows:
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.
Now, if has dimension . We can total order bases and . Then a change of basis (from to ) has the matrix representation
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.
Since is obviously invertible, is invertible also, whose inverse is . Furthermore, for any basis . Here, is the identity matrix.
Examples.
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1.
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.
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2.
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:
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3.
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 .
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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 .
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This also means that every invertible matrix corresponds to (in a one-to-one fashion) a change of basis from the basis whose elements are columns of to , the standard basis: .
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Continue to assume that is the standard basis. Let be any ordered bases for . Using the above property, we can easily compute , which is
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Let be a re-ordering of the ordered basis , where each is just for some permutation in . Then is the permutation matrix corresponding to the permutation .
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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
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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.
References
- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
Title | change of basis |
Canonical name | ChangeOfBasis |
Date of creation | 2013-03-22 17:30:18 |
Last modified on | 2013-03-22 17:30:18 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 20 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 15A04 |
Synonym | change of coordinates |
Synonym | change of bases |
Synonym | basis change |
Synonym | base change |
Synonym | base change matrix |
Defines | change of basis matrix |