Math 205a Quiz 03 Worksheet - 2008
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1Math 205A
Quiz 03 page 1
September 26, 2008
NAME
8
4
0
0
1
0
1. Suppose
x
+ x
, where x
and x
are free, is the solution v
of the homogeneous
3
5
3
5
h
0
3
0
1
0
0
matrix equation Bx = 0 for some matrix B. Also, let v
, v
, . . . , be the column vectors of B.
1
2
1A. You can not tell from the above info how many rows B has. But how many columns must B have,
and how do you know?
1B. Is the set {v
, v
, . . . } of column vectors of B linearly independent? Explain in terms of the
1
2
definition of LI (that is, consider the solutions of x
v
+ x
v
+ · · · = 0).
1
1
2
2
1C. Use the equation x
v
+ x
v
+ · · · = 0 to express v
as a specific linear combination of the other
1
1
2
2
1
column vectors, or explain why this is impossible.
1D. Express v
as a linear combination of the other column vectors, or explain why this is impossible.
2
1E. Let b = 7v
+ 6v
12v
in the following two questions:
1
2
4
1E (i). Express b as a linear combination of the column vectors of B without using v
(by replacing v
4
4
with a LC of the other column vectors).
1E (ii). Can you express b as a linear combination of the column vectors of B without using v
? Explain
2
your answer.
1 Challenge Bonus Question: Let m be the number of rows of B. Suppose Bx = c does not have a
m
solution for every c in R
. What is the RREF of B, where m is as small as possible?
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