MAS 213: Linear Algebra II.
Problem set #11.
Solutions
.
Problem 1:
(Problem 6.2.2 from [FIS], various parts.)
In the following parts you are given an inner product space
V
, a set of vectors
S
⊂
V
, and a distinguished vector
x
. In each case find an orthonormal basis
for span(
S
), then use the inner product to compute the co-ordinate vector of
the given vector
x
with respect to the basis that you have just determined.
Check your answer.
Solution to part (i).
In this part the vector space is
R
3
with the standard inner product, the given
set of vectors is
S
=
{
(1
,
0
,
1)
,
(0
,
1
,
1)
,
(1
,
3
,
3)
}
,
and the distinguished vector is
x
= (1
,
1
,
2).
This part is easier than it looks.
Note that the vectors in this set are
linearly independent, so, because there are three of them, their span is all of
R
3
.
We are asked to find an orthonormal basis for their span, which is
R
3
, so
we might as well take the standard orthonormal basis
β
=
{
(1
,
0
,
0)
,
(0
,
1
,
0)
,
(0
,
0
,
1)
}
.
The co-ordinate vector of
x
= (1
,
1
,
2) with respect to this basis is obvi-
ously itself. The inner product calculation gives that answer too:
x
=
(
x
·
(1
,
0
,
0))(1
,
0
,
0) + (
x
·
(0
,
1
,
0))(0
,
1
,
0) + (
x
·
(0
,
0
,
1))(0
,
0
,
1)
=
(1)
∗
(1
,
1
,
2) + (1)
∗
(0
,
1
,
0) + (2)
∗
(0
,
0
,
1)
.
1