## Description

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