Change of Basis.
Changing basis is basically (pun intended) what stage II algebra is all about. Row reduction and column reduction are just fancy ways of changing basis. The process of diagonalising a matrix is just choosing a basis in which the matrix looks very nice.
It is very important at this level that the process of changing from one basis to another is completely natural and well understood. These worksheets are intended to ensure that this is so. The idea is that if everyone works through them at their own pace, by the end of it all everyone will be able to change basis as easily as a musician changes key.
We will be working with the vector space V =R3. In this vector space we are used to working with the usual basis
U ={e1,e2,e3}
If v is an element of V, then we know how to write v as a column vector with respect to the basis U. If v= c1e1+c2e2+c3e3 then we write v as a column vector
v=
c1 c2 c3
Suppose now that we have a different basis of V =R3, namely B={v1,v2,v3}
Then we can also write v in terms of the basis B. If v=a1v1+a2v2+a3v3 then we write v as a column vector
v=
a1 a2 a3
B
where the subscriptBis there to remind us that this vector is not written in terms of the usual basis, but is written in terms of the basis B.
1
In fact we can define the expression
a1 a2 a3
B
as simply an abbreviation for the expression a1v1+a2v2+a3v3, where B is the ordered basis{v1,v2,v3}. When we are talking about the usual basisU, we sometimes get lazy and don’t bother to write the basis down.
There is an alternative notation that is best for theoretical work. In this alternative notation we attach the basis label in a different place, and write
[v]B =
a1 a2 a3
The right hand side here is to be interpreted as just a box of numbers.
Someone needs to tell you what basis is being used before you can interpret it as a vector in V. The left hand side of the equation could be translated as“the box of numbers obtained by writing vector vas a column vector using basis B”. For practical computations this notation is clumsy, however it is much better for theoretical work. We will be using the first notation for the rest of this handout.
To sum up we now have three entirely equivalent ways to write down the same fact.
1. v=a1v1 +a2v2+a3v3
2. v=
a1 a2 a3
B 3. [v]B =
a1
a2 a3
OK. Now it is time to get into some computations. We define the following different bases of V.
2
U , the usual basis B =
1 0 2
,
2 1 0
,
0 1 2
C =
1 1 1
,
1 1 2
,
1 2 2
Remember that if I don’t specify a basis for a vector then the convention is that I am using the usual basis.
Exercise 1: Express each of the following vectors in terms of each of the bases above.
1 0 4
B ,
1 1 1
C ,
3 3 3
,
4
−2
−2
C
You probably noticed that I did not state a particular method for solving the problems in this exercise. This is because I want you to get to grips with what change of basis really is, rather than just applying some rote method that you have learned.
You will have found that some of the column vectors were very easy to find. Others were not so easy. It is pretty easy to turn a column vector in basis B into a column vector in the usual basis, for example, since we have expressions for the elements ofBin terms of the usual basis. It is not so easy to go the other way.
For example,
v=
a b c
B
=a
1 0 2
+b
2 1 0
+c
0 1 2
=
1.a+ 2.b+ 0.c 0.a+ 1.b+ 1.c 2.a+ 0.b+ 2.c
U so converting from basis B to basisU is as easy as multiplying by a matrix.
This matrix, whose columns are the elements of B expressed in the basis U, is called the change of basis matrix from B to U. To remind us that it
3
should multiply a vector in basis B, and gives an answer which should be interpreted using the basis U, we write this matrix as
1 2 0 0 1 1 2 0 2
U B Now we can write
v=
a b c
B
=a
1 0 2
U +b
2 1 0
U +c
0 1 2
U
=
1 2 0 0 1 1 2 0 2
U B
a b c
B This equation sums up the whole process of changing basis very nicely.
Exercise 2: There are six different ways to convert amongst the basesU,B, and C. This gives us six different change of basis matrices. We wrote down one of them above. Work out the other five. To do this you will have to work out what vectors in one basis look like in another basis.
This will give you the columns of your matrix.
Exercise 3: These matrices are related to one another. Show that the ma- trix for changing from U toB is the inverse of the matrix for changing from B to U. Think about what products of these matrices do to the elements of V. Does this make sense?
Also show that the matrix for changing from B to C is the product of the matrices which change fromB toU, and from U toC. Which way around does the product go? Does this make sense in terms of what these matrices do to the elements of V?
Get familiar with the process of using change of basis matrices to convert vectors from one basis into another. Coming soon to a (lecture) theatre near you will be the next handout in this series which will address the topic of changing matrices from one basis to another.
4
