$30.00
1. Let ⃗u =
3
4
1
, ⃗v =
4
−4
−4
, and ⃗w =
14
0
d
.
(a) For what value(s) of d is span{⃗u, ⃗v, ⃗w} a plane?
(b) Is there a value of d so span{⃗u, ⃗v, ⃗w} a line? Explain.
2. Let G ⊆ R
2 be the graph of the line given by the equation y = 3x + 2. Find vectors ⃗d and ⃗p
so that
G = {⃗x ∈ R
2
: ⃗x = t
⃗d + ⃗p for some t ∈ R}.
3. Let ⃗v =
[
1
2
]
,
X = {⃗x ∈ R
2
: ⃗x · ⃗v = 0}
Y = {⃗x ∈ R
2
: ⃗x · ⃗v = −3}.
(a) Draw X and Y . What do you notice? Are either of them subspaces?
(b) Find a vector ⃗w so that
Y = {⃗x + ⃗w : ⃗x ∈ X}.
4. Let ⃗x =
[
1
1
]
and ⃗y =
[
1
−1
]
. Prove that {⃗x, ⃗y} is a basis for R
2
.
5. Let
U = span
1
2
3
,
3
2
1
,
4
4
4
,
V =
⃗x ∈ R
3
: ⃗x ·
1
1
1
= 0
,
W = U ∪ V.
For each subset U, V , W of R
3
, show whether it is a subspace or not. If it is a subspace,
classify it as a point, line, plane, or all of R
3
. Further, if it is a subspace, give a basis for it.
1