Solutions To Linear Algebra Practice Problems Page 9
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9. Consider the following basis for
:
1
3
E =
,
2
5
2
(a) Find the coordinates for the vector
in terms of the basis E.
4
2
2
(b) Let L :
be the following linear transformation:
L(x, y) = (2x
y, 3x
2y)
Find the matrix representing L with respect to the basis E.
Answer:
(a) We need to find numbers c
and c
such that
1
2
1
3
2
c
+ c
=
1
2
2
5
4
Thus, we need to solve the following system of linear equations:
c
+ 3c
=
2
1
2
2c
+ 5c
= 4
1
2
Solving, we get c
= 22 and c
=
8. Thus, the coordinates in
1
2
22
terms of basis E are
8
(b) First, we apply L to each of the basis vectors:
1
0
L
=
2
1
3
1
L
=
5
1
Then, we find the coordinates for each of the resulting vectors in
terms of the basis E. First, we compute the transition matrix to
change from the standard basis to the basis E:
1
1
3
5
3
=
2
5
2
1
9
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