Math 330 Homework 5.4 (Pages 333,334)
(4)
Let
B
=
{
vector
b
1
,
vector
b
2
,
vector
b
3
}
be a basis for a vector space
V
and
T
:
V
→
R
2
a linear transformation
with the property that
T
(
x
1
vector
b
1
+
x
2
vector
b
2
+
x
3
vector
b
3
) =
bracketleftBigg
2
x
1

4
x
2
+ 5
x
3

x
2
+ 3
x
3
bracketrightBigg
Find the matrix for
T
relative to
B
and the standard basis for
R
2
SOLUTION:
Our matrix with respect to the two bases is essentially a table where the rows describe the
output basis for
R
2
and the columns the input basis for
V
.
In what appears below we let
vector
epsilon1
1
=
bracketleftBigg
1
0
bracketrightBigg
and
vector
epsilon1
2
=
bracketleftBigg
0
1
bracketrightBigg
denote the two elements for the standard basis of
R
2
.
T
vector
b
1
vector
b
2
vector
b
3
vector
epsilon1
1
2
4
5
vector
epsilon1
2
0
1
3
The table describes the following facts in a very compact way:
•
With
x
1
= 1 and
x
2
=
x
3
= 0 we have that
T
(
vector
b
1
) = 2
vector
epsilon1
1
+ 0
vector
epsilon1
2
=
bracketleftBigg
2
0
bracketrightBigg
.
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 Spring '08
 Johnson,J
 Linear Algebra, Algebra, Vector Space, basis, linear transformation

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