Name (print): __________________
Student Number:
__________________
Test #2 ( /30)
Part A: Multiple Choice (1 Mark Each, No Penalty For Incorrect Answers)
Question 1: Which of the following sets is linearly independent:
0 1 0
3 6
0 1 0
1
a)
,
b)
,
,
c
)
,
,
,
d) All are linearly dependent
1 2
0 0 1
1 0 1
1
0 0 0
2 4
1 0 0
2
Question 2: The dimension of is
a)
3
b)
4 c)
5
d) ∞ e) None of the above
Part B: Fill in the Blank (1 Mark Each)
Question 1: Give the standard basis for :
1 0 0 0
0 , 1 , 0 , 0
0 0 1 0
0 0 0 1
Question 2: Give the definition of a span of vectors , ,
The span of , , is the set of all linear combinations of
where , , can be any
real number.
Part C: Short Answer (Show all work)
,
Question 1: Consider a space with the following operations:
a)
What would be the scalar r that would make
0 as
0
0
gives: 0
[1 Mark]
b)
What would be the additive inverse vector for this space for the vector
[1 Mark]
The additive inverse for
c)
would be
0
0
as
Show that this space fails (u + v) + w = u + (v + w) by giving an example that will not make them equal:
[2 Marks]
Choose
1
,
1
1
,
0
0
then
1
1
1
[1 Mark for the left side calculation]
[1 Mark for the right side calculation]
1
1
1
0
1
0
0
1
0
1
1
1
2
1
0
1
1
1
2
0
2
2
Question 2: Show that the following is a subspace of
[6 Marks]
|
0
1. Clearly this is of proper form (it has three entries written as a column vector just like )
0
2. The zero vector in is 0 , we can get this by making a = b = c = 0. This satisfies a+b=0+0=0 needed in S.
0
3. For the addition property: let
,
...