Math 205a Quiz 03 Worksheet - Bates College - 2007
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1Math 205A
Quiz 03, page 1
September 28, 2007
NAME
m
1. Let S be the set of vectors {c
c
, . . . , c
}, where each c
is a column vector in R
. Some of
1
2
k
i
the following statements are true (ie, they are theorems) and others are not. Circle each letter for
which the statement is true, and mark an “X” through the letter if the statement is false.
a. If m < k then S is linearly dependent.
b. If k < m then S is linearly independent.
c. If S contains the zero vector, S is linearly dependent.
d. If S is linearly dependent, then every vector in S can be written as a linear combination of
the other vectors in S.
e. If at least one vector in S is not a linear combination of other vectors in S, then S is linearly
independent.
f. The set S is linearly dependent ⇐⇒ at least one vector in S can be expressed as a linear
combination of the others.
g. If the only solution to Ax = 0 is the trivial solution, where the members of S form the columns
of A, then S is linearly independent.
2. By inspection, classify each of the following sets as linearly independent (LI) or linearly
dependent (LD). Justify your answer by writing the letter corresponding to the theorem from
problem 1 next to your answer of “LI” or “LD”. For example, if you think the set in 2Q is linearly
independent and statement j in problem 1 is your reason, then you’d write “LI-j” as your answer.
1
2
3
4
1
2
1
0
0
2
3
4
,
,
,
2A.
6
7
0
0
2B.
,
,
,
0
0
3
4
7
8
0
1
0
0
0
4
7
5
0
12
2
e
+ 3
sin 2π
3
5
0
2
2C.
,
2D.
,
,
,
0
cos π/4
e
1
2
5
1
7
4
5
4
9
1 0 2 3
3. Suppose the matrix A has column vectors c
, c
, c
, c
where A is:
0 1 0 0
1
2
3
4
3 0 5 8
3A: Express c
as a linear combination of other column vectors or explain why you can’t.
1
3B: Express c
as a linear combination of other column vectors or explain why you can’t.
2
3C: This example should explain why one of the statements in problem 1 above is false. Which
one?
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