You can navigate through this E-Book using the menu to the left. For E-Books that have a Chapter/Section organization each option in the menu to the left indicates a chapter and will open a menu showing the sections in that chapter. Alternatively, you can navigate to the next/previous section or chapter by clicking the links in the boxes at the very top and bottom of the material.
Also, depending upon the E-Book, it will be possible to download the complete E-Book, the chapter containing the current section and/or the current section. You can do this be clicking on the E-Book, Chapter, and/or the Section link provided below.
For those pages with mathematics on them you can, in most cases, enlarge the mathematics portion by clicking on the equation. Click the enlarged version to hide it.
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.
|
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]
|
