Solutions to HW #4
Math 67
UC Davis, Fall 2011
1. Solve the following problems in the textbook:
(a) Proofwriting exercise 6 in Chapter 5.
Solution.
Since
U
+
V
⊂
R
9
, we have dim(
U
+
V
)
≤
9, and therefore, by
theorem 5.4.6 in the textbook,
dim(
U
∩
V
) = dim(
U
)+dim(
V
)

dim(
U
+
V
)
≥
dim(
U
)+dim(
V
)

9 = 5+5

9 = 1
.
So
U
∩
V
must be larger than the 0dimensional space
{
0
}
.
(b) Calculational exercises 1(a),(b),(c),(f), 2(a)–(b), 5, 6 in Chapter 6.
Solution to 1(b),(c).
For each (
a,b
)
∈
R
2
, it is easy to ﬁnd that the equation
(
x
+
y,x
) =
T
(
x,y
) = (
a,b
) has the unique solution (
x,y
) = (
b,a

b
). Since there
is a solution, that means the transformation is surjective. Since the solution is
unique, that means the transformation is injective, which as we saw is equivalent
to null(
T
) =
{
0
}
, so dim(null(
T
)) = 0.
Solution to 1(f).
F
(0
,
0) = (0
,
1). Since
F
does not map the zero vector in the
domain to the zero vector in the codomain, it is not a linear transformation.
(c) Proofwriting exercises 2, 3 in Chapter 6.