5.3. Coordinates and change of basis in a vector space 





(i) Let
= {}
be the natural of .
Then for any v >(x, y, z)
,
v =
.
Thus,
Similar statements holds for
.
(ii)
Let V =
,
the vector space of all real polynomial of degree at most 3. Let
= {},
.
is the standard basis for
.
It is early to check that
is also basis for
.
For any vector p(x)
=
,
Note that,
=
if and only if
a =
b =

c = 2
+
+
Hence,
= a,
= (),
= ().
Thus for p(x)
=
,
An element
v
V uniquely determines and is determined by its coordinate
vector. In fact, we have the following : 

















Let V =
,
=
{(2, 0, 1), (1, 2, 0), (1, 1, 1)} and
=
.
We ask the reader to verify that both
and
are bases for .
Let v = (4, 9, 5)
.
Let us first complete
and
.
Let v =
,
i.e.,
It is early to see that this implies
= 4,
= 5,
= 1. Similarly, if v =
,
i.e.,
then a = 1, b = 2, c = 3. Hence
Next to find the matrix P such that
We recall that the
column of P is the coordinate vector
if
with respect to the basis
.
This means we have to solve the equations
Note that each is a system of linear equations:
Since then system has same coefficient matrix, we can show these
together. Consider the augmented matrix
to reduced rowechelon form. This (we leave the calculation to the
reader) is
Thus
Using (5.2) and (5.4), it is easy to check that equation (5.3)
holds.
Similarly, the matrix Q that hold gives the change of conditions
from
to
,
i.e.,
,
can be computed by solving.
to get
Finally, we ask the reader to verify that PQ = QP =
Id. (iii) Let
={(1, 3), (2, 4)} be a basis of
and let
Let us find a basis
of
such that the matrix for the change of basis from
to
is P. For this, we note that
Hence,
= {(7, 9), (5, 7)}. 


Click
here to see an interactive visualization: Applet 5.3 
Click here to take a Quiz: Quiz 5.3 





