Math 205b Quiz 02 - University Of California, 2009
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1Quiz 02 page 1
09/25/2009
Math 205B
Name
m
1. Let b and v
, v
, . . . , v
be vectors in R
. Complete the following sentence so that it gives the definition of linear
1
2
n
“We say b is a linear combination of the the vectors v
, v
, . . . , v
if and only if. . . ”
combination:
1
2
n
2
6
1
10
16
26
1
3
0
4
8
8
2. Let a
=
, a
=
, a
=
. a
=
. Also, let b =
and c =
.
1
2
3
4
3
9
4
4
26
16
2A. Is b in the span of {a
}? Explain your answer. Show any matrices and corresponding rref’s you use.
, a
, a
, a
1
2
3
4
2B. Let A be the matrix whose columns are a
, a
, a
and a
. Express all solutions of Ax = c in parametric vector
1
2
3
4
form, that is, as p + v
where p is a particluar solution of Ax = c and v
is all solutions of the corresponding homogeneous
h
h
equation Ax = 0. Show any relevant matrices used in your work.
2C. Use your work in (2B) to find two nontrivial solutions s
and s
of Ax = 0. CIRCLE your answers.
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2
2D. Now let T be the matrix whose columns are a
, a
and a
(so T looks like A if you take out A’s second column).
1
3
4
What is v
now? That is, what are the solutions of T x = 0?
h
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