Math 2040 Matrix Theory and Linear Algebra II sample midterm ...

[ ]
[ ]
r1
x1
(e) (Abstract vector spaces.) Let V = { | r
r 1 +r 2 = 1 and r 1 , r 2 ∈ R}. For any ∈ V ,
2 x
[ ]
[ ] [ ]
[ 2
]
y1
x1 y1
x1 + y
∈ V , and α ∈ R, suppose + in V is defined to be
1 − 1
and
y 2 x 2 y 2 x 2 + y
[ ]
[ ]
2
x1
αx1 + (1 − α)
α in V is defined to be
. It turns out that V is a vector space with
x 2 αx 2
this addition and scalar multiplication defined on it. Answer the following four questions
about the vector space V .
i. What is 0 ∈ V [
]
1
Solution: 0 = in V .
0
[ ]
4
ii. What is the additive inverse of in V
−3
[ ]
[ ] [ ]
−2 4
1
Solution: , since adding this to results in .
3
−3
0
[ ]
4
iii. What is 5 in V
−3
[ ] [ ]
5(4) + (1 − 5)
16
Solution:
; that is, .
5(−3)
−15
iv. V is a subset of R 2 , there is a 0 ∈ V , and the given addition and multiplication by
scalars are closed operations on V . However, V a not subspace of R 2 with this vector
space structure defined on [ V . Why ]
0
Solution: The 0 ∈ R 2 is , but this is not in V . Also, the addition and scalar
0
multiplication given for V does not correspond to addition and scalar multiplication
inherited from R 2 .
2. Representations. (Value = 40%)
Let B = {b 1 , b 2 , b 3 } be a basis of a vector space V . Let
c 1 = b 1 + b 2 ,
c 2 = b 2 + b 3 , and
c 3 = b 1 + b 3 .
Let C = {c 1 , c 2 , c 3 }.
(a) (Coordinates.) If [x] B =
⎡
⎣ −1 1
−2
⎤
⎦, then what is x
Solution: x = −b 1 + b 2 − 2b 3
2