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In Example 1 of the
previous section we saw that the vectors  ,
and formed a basis for . This means that every vector in ,
for example the vector ,
can be written as a linear combination of these three vectors. Of course this is not the only basis for . There are many other bases for out there in the world, not the least of which
is the standard basis for ,
The standard basis for any vector space is generally the
easiest to work with, but unfortunately there are times when we need to work
with other bases. In this section we’re
going to take a look at a way to move between two different bases for a vector
space and see how to write a general vector as a linear combination of the
vectors from each basis.
To start this section off we’re going to first need a way to
quickly distinguish between the various linear combinations we get from each
basis. The following definition will
help with this.
Note that by Theorem 1
of the previous section we know that the linear combination of vectors from the
basis will be unique for u and so
the coordinate vector will also be unique.
Also, on occasion it will be convenient to think of the
coordinate vector as a matrix. In these
cases we will call it the coordinate
matrix of u relative to S.
The coordinate matrix will be denoted and defined as follows,
At this point we should probably also give a quick warning
about the coordinate vectors. In most
cases, although not all as we’ll see shortly, the coordinate vector/matrix is
NOT the vector itself that we’re after.
It is nothing more than the coefficients of the basis vectors that we
need in order to write the given vector as a linear combination of the basis
vectors. It is very easy to confuse the
coordinate vector/matrix with the vector itself if we aren’t paying attention,
so be careful.
Let’s see some examples of coordinate vectors.
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Example 1 Determine
the coordinate vector of relative to the following bases.
(a) The
standard basis vectors for ,
. [Solution]
(b) The
basis where, ,
and . [Solution]
Solution
In each case we’ll need to determine who to write as a linear combination of the given basis
vectors.
(a) The standard basis
vectors for , .
In this case the linear combination is simple to write
down.
and so the
coordinate vectors for x relative
to the standard basis vectors for is,
So, in the case
of the standard basis vectors we’ve got that,
this is, of
course, what makes the standard basis vectors so nice to work with. The coordinate vectors relative to the
standard basis vectors is just the vector itself.
[Return to Problems]
(b)
The basis where, , and .
Now, in this
case we’ll have a little work to do.
We’ll first need to set up the following vector equation,
and we’ll need
to determine the scalars ,
and . We saw how to solve this kind of vector
equation in both the section on Span and the section on Linear
Independence. We need to set up the
following system of equations,
We’ll leave it
to you to verify that the solution to this system is,
The coordinate
vector for x relative to A is then,
[Return to Problems]
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As always we should do an example or two in a vector space
other than .
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Example 2 Determine
the coordinate vector of relative to the following bases.
(a) The
standard basis for . [Solution]
(b) The
basis for ,
,
where ,
,
and . [Solution]
Solution
(a) The standard basis for
.
So, we need to write p
as a linear combination of the standard basis vectors in this case. However, it’s already written in that
way. So, the coordinate vector for p relative to the standard basis
vectors is,
The ease with which we can write down this vector is why
this set of vectors is standard basis vectors for .
[Return to Problems]
(b) The basis for , , where , , and .
Okay, this set is similar to the standard basis vectors,
but they are a little different so we can expect the coordinate vector to
change. Note as well that we proved in
Example 5(b) of the previous section
that this set is a basis.
We’ll need to find scalars , and for the following linear combination.
The will mean
solving the following system of equations.
This is not a
terribly difficult system to solve.
Here is the solution,
The coordinate
vector for p relative to this
basis is then,
[Return to Problems]
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Example 3 Determine
the coordinate vector of relative to the following bases.
(a) The
standard basis of ,
. [Solution]
(b) The
basis for ,
where ,
,
,
and . [Solution]
Solution
(a) The standard basis of , .
As with the previous two examples the standard basis is
called that for a reason. It is very
easy to write any matrix as a linear combination of these
vectors. Here it is for this case.
The coordinate
vector for v relative to the
standard basis is then,
[Return to Problems]
(b)
The basis for , where , , , and .
This one will
be a little work, as usual, but won’t be too bad. We’ll need to find scalars ,
,
and for the following linear combination.
Adding the
matrices on the right into a single matrix and setting components equal gives
the following system of equations that will need to be solved.
Not a bad
system to solve. Here is the solution.
The coordinate
vector for v relative to this
basis is then,
[Return to Problems]
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Before we move on we should point out that the order in
which we list our basis elements is important, to see this let’s take a look at
the following example.
Now that we’ve got the coordinate vectors out of the way we
want to find a quick and easy way to convert between the coordinate vectors
from one basis to a different basis.
This is called a change of basis.
Actually, it will be easier to convert the coordinate matrix for a
vector, but these are essentially the same thing as the coordinate vectors so
if we can convert one we can convert the other.
We will develop the method for vectors in a 2-dimensional
space (not necessarily ) and in the process we will see how to
do this for any vector space. So let’s
start off and assume that V is a
vector space and that . Let’s also suppose that we have two bases for
V.
The “old” basis,
and the “new” basis,
Now, because B is
a basis for V we can write each of
the basis vectors from C as a linear
combination of the vectors from B.
This means that the coordinate matrices of the vectors from C relative to the basis B are,
Next, let u be
any vector in V. In terms of the new basis, C, we can write u as,
and so its coordinate matrix relative to C is,
Now, we know how to write the basis vectors from C as linear combinations of the basis
vectors from B so substitute these
into the linear combination for u
above. This gives,
Rearranging gives the following equation.
We now know the coordinate matrix of u is relative to the “old” basis B. Namely,
We can now do a little rewrite as follows,
So, if we define P
to be the matrix,
where the columns of P
are the coordinate matrices for the basis vectors of C relative to B, we can
convert the coordinate matrix for u
relative to the new basis C into a
coordinate matrix for u relative to
the old basis B as follows,
Note that this may seem a little backwards at this
point. We’re converting to a new basis C and yet we’ve found a way to instead
find the coordinate matrix for u
relative to the old basis B and not
the other way around. However, as we’ll
see we can use this process to go the other way around. Also, it could be that we have a coordinate
matrix for a vector relative to the new basis and we need to determine what the
coordinate matrix relative to the old basis will be and this will allow us to
do that.
Here is the formal definition of how to perform a change of
basis between two basis sets.
We should probably take a look at an example or two at this
point.
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Example 5 Consider
the standard basis for ,
,
and the basis where, ,
and .
(a) Find
the transition matrix from C to B.
[Solution]
(b) Find
the transition matrix from B to C.
[Solution]
(c) Use
the result of part (a) to compute given . [Solution]
(d) Use
the result of part (a) to compute given . [Solution]
(e) Use
the result of part (b) to compute given . [Solution]
(f) Use
the result of part (b) to compute given . [Solution]
Solution
Note as well that we gave the coordinate vector in the last
four parts of the problem statement to conserve on space. When we go to work with them we’ll need to
convert to them to a coordinate matrix.
(a) Find the transition
matrix from C to B.
When the basis we’re going to (B in this case) is the standard basis vectors for the vector
space computing the transition matrix is generally pretty simple. Recall that the columns of P are just the coordinate matrices of
the vectors in C relative to B.
However, when B is the
standard basis vectors we saw in Example 1 above that the coordinate vector
(and hence the coordinate matrix) is simply the vector itself. Therefore, the coordinate matrix in this
case is,
[Return to Problems]
(b)
Find the transition matrix from B to C.
First, do not
make the mistake of thinking that the transition matrix here will be the same
as the transition matrix from part (a). It won’t be. To find this transition matrix we need the
coordinate matrices of the standard basis vectors relative to C.
This means that we need to write each of the standard basis vectors as
linear combinations of the basis vectors from C. We will leave it to you
to verify the following linear combinations.
The coordinate
matrices for each of this is then,
The transition
matrix from C to B is then,
So, a
significantly different matrix as suggested at the start of this problem. Also, notice we used a slightly different
notation for the transition matrix to make sure that we can keep the two
transition matrices separate for this problem.
[Return to Problems]
(c)
Use the result of part (a) to compute given .
Okay, we’ve done most of the work for this
problem. The remaining steps are just
doing some matrix multiplication. Note
as well that we already know what the answer to this is from Example 1
above. Here is the matrix
multiplication for this part.
Sure enough we
got the coordinate matrix for the point that we converted to get from Example 1.
[Return to Problems]
(d)
Use the result of part (a) to compute given .
The matrix
multiplication for this part is,
So, what have
we learned here? Well, we were given
the coordinate vector of a point relative to C. Since the vectors in C are not the standard basis vectors
we don’t really have a frame of reference for what this vector might actually
look like. However, with this
computation we know now the coordinates of the vectors relative to the standard
basis vectors and this means that we actually know what the vector is. In this case the vector is,
So, as you can
see, even though we’re considering C
to be the “new” basis here, we really did need to determine the coordinate
matrix of the vector relative to the “old’ basis here since that allowed us
to quickly determine just what the vector was. Remember that the coordinate matrix/vector
is not the vector itself, only the coefficients for the linear combination of
the basis vectors.
[Return to Problems]
(e)
Use the result of part (b) to compute given .
Again, here we
are really just verifying the result of Example 1 in this part. Here is the matrix multiplication for this
part.
And again, we
got the result that we would expect to get.
[Return to Problems]
(f)
Use the result of part (b) to compute given .
Here is the
matrix multiplication for this part.
So what does
this give us? We’ll first we know that
. Also, since B is the standard basis vectors we know that the vector from that we’re starting with is . Recall that when dealing with the standard
basis vectors for the coordinate matrix/vector just also
happens to be the vector itself.
Again, do not always expect this to happen.
The coordinate
matrix/vector that we just found tells us how to write the vector as a linear
combination of vectors from the basis C. Doing this gives,
[Return to Problems]
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Example 6 Consider
the standard basis for ,
and the basis ,
where ,
,
and .
(a) Find
the transition matrix from C to B.
[Solution]
(b) Determine
the polynomial that has the coordinate vector . [Solution]
Solution
(a) Find the transition
matrix from C to B.
Now, since B
(the matrix we’re going to) is the standard basis vectors writing down the
transition matrix will be easy this time.
Each column of P will be the coefficients of the
vectors from C since those will
also be the coordinates of each of those vectors relative to the standard
basis vectors. The first row will be
the constant terms from each basis vector, the second row will be the
coefficient of x from each basis
vector and the third column will be the coefficient of from each basis vector.
[Return to Problems]
(b) Determine the
polynomial that has the coordinate vector .
We know what the coordinates of the polynomial are
relative to C, but this is not the
standard basis and so it is not really clear just what the polynomial
is. One way to get the solution is to
just form up the linear combination with the coordinates as the scalars in
the linear combination and compute it.
However, it would be somewhat illustrative to use the
transition matrix to answer this question.
So, we need to find and luckily we’ve got the correct transition
matrix to do that for us. All we need
to do is to do the following matrix multiplication.
So, the
coordinate vector for u relative
to the standard basis vectors is
Therefore, the polynomial
is,
Note that, as
mentioned above we can also do this problem as follows,
The same answer
with less work, but it won’t always be less work to do it this way. We just wanted to point out the alternate
method of working this problem.
[Return to Problems]
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Example 7 Consider
the standard basis for ,
,
and the basis where ,
,
,
and .
(a) Find
the transition matrix from C to B.
[Solution]
(b) Determine
the matrix that has the coordinate vector . [Solution]
Solution
(a) Find the transition
matrix from C to B.
Now, as with the previous couple of problems, B is the standard basis vectors but
this time let’s be a little careful.
Let’s find one of the columns of the transition matrix in detail to
make sure we can quickly write down the remaining columns. Let’s look at the fourth column. To find this we need to write as a linear combination of the standard
basis vectors. This is fairly simple
to do.
So, the
coordinate matrix for relative to B and hence the fourth column of P is,
So, each column
will be the entries from the ’s and with the standard basis
vectors in the order that we’ve using them here, the first two entries is the
first column of the and the last two entries will be the second
column of . Here is the transition matrix for this
problem.
[Return to Problems]
(b)
Determine the matrix that has the coordinate
vector .
So, just as
with the previous problem we have the coordinate vector, but that is for the
non-standard basis vectors and so it’s not readily apparent what the matrix
will be. As with the previous problem
we could just write down the linear combination of the vectors from C and compute it directly, but let’s
go ahead and used the transition matrix.
Now that we’ve
got the coordinates for v relative
to the standard basis we can write down v.
[Return to Problems]
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To this point we’ve only worked examples where one of the
bases was the standard basis vectors.
Let’s work one more example and this time we’ll avoid the standard basis
vectors. In this example we’ll just find
the transition matrices.
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Example 8 Consider
the two bases for ,
and .
(a) Find
the transition matrix from C to B.
[Solution]
(b) Find
the transition matrix from B to C.
[Solution]
Solution
Note that you should verify for yourself that these two
sets of vectors really are bases for as we claimed them to be.
(a) Find the transition
matrix from C to B.
To do this we’ll need to write the vectors from C as linear combinations of the
vectors from B. Here are those linear combinations.
The two
coordinate matrices are then,
and the
transition matrix is then,
[Return to Problems]
(b)
Find the transition matrix from B to C.
Okay, we’ll
need to do pretty much the same thing here only this time we need to write
the vectors from B as linear
combinations of the vectors from C. Here are the linear combinations.
The coordinate
matrices are,
The transition
matrix is,
[Return to Problems]
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In Examples 5 and 8 above we computed both transition
matrices for each direction. There is
another way of computing the second transition matrix from the first and we will
close out this section with the theorem that tells us how to do that.
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Theorem 1 Suppose
that V is a finite dimensional
vector space and that P is the
transition matrix from C to B then,
(a) P is invertible and,
(b) is the transition matrix from B to C.
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You should go back to Examples 5 and 8 above and verify that
the two transition matrices are in fact inverses of each other. Also, note that due to the difficulties
sometimes present in finding the inverse of a matrix it might actually be easier
to compute the second transition matrix as we did above.
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