Name:
MAS 3114 Computational Linear Algebra
Fall 1997
FINAL EXAM
Instructions:
•
There are 12 questions.
•
There 223 points available. Full marks (100%) will be given for 210 points.
•
Write on
ONE
side of the paper provided.
•
Show all necessary working and reasoning to receive full credit.
•
Your work needs to be written in a proper and coherent fashion.
•
When giving proofs your reasoning should be clear.
•
Calculators are allowed (for simple calculations).
•
Questions with
may be answered on the exam paper.
1
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1.
[4 + 10 + 4 = 18 points]
(a) (i)
Let
v
1
,
v
2
, . . . ,
v
p
be vectors in
R
n
.
Define the set denoted by Span
{
v
1
,
v
2
, . . . ,
v
p
}
.
(b) Let
v
1
=
1
4

2
,
v
2
=
5
19
4
,
v
3
=
3
8

19
. Let
W
= Span
{
v
1
,
v
2
}
.
(i) Determine if
v
3
is in the subspace
W
, showing working. Use PART 1 of the MATLAB
HANDOUT to check your answer.
(ii) Find dim
W
giving reason.
2.
[4 points]
Complete the following
definition
: A set
{
v
1
,
v
2
, . . . ,
v
p
}
in
R
n
is said to be
linearly independent
if the vector equation
has
.
3.
[5
×
4 = 20 points]
Let
v
1
=
1

3

3
,
v
2
=

1
4
1
,
v
3
=
0
1

2
,
v
4
=
0
0
0
,
v
5
=
1
2
3
,
v
6
=
v
1

2
v
3
,
v
7
=
3
2
1
.
Determine which of the following sets of vectors are linearly independent giving reasons. You might
find PART 2 of the MATLAB HANDOUT useful.
(i)
{
v
1
,
v
2
,
v
3
}
,
(ii)
{
v
2
,
v
3
,
v
4
}
,
(iii)
{
v
3
,
v
5
}
,
(iv)
{
v
1
,
v
3
,
v
5
,
v
7
}
,
(v)
{
v
1
,
v
3
,
v
6
}
.
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 Fall '08
 Olson
 Linear Algebra, Matlab Handout

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