Math 205A Quiz 03 page 1 September 26, 2008 NAME
1. Suppose x3
−8
0
1
0
0
0
+x5
4
0
0
−3
1
0
, where x3and x5are free, is the solution vhof the homogeneous
matrix equation Bx=0for some matrix B. Also, let v1,v2, . . . , be the column vectors of B.
1A. You can not tell from the above info how many rows Bhas. But how many columns must Bhave,
and how do you know?
1B. Is the set {v1,v2,...}of column vectors of Blinearly independent? Explain in terms of the
definition of LI (that is, consider the solutions of x1v1+x2v2+··· =0).
1C. Use the equation x1v1+x2v2+··· =0to express v1as a specific linear combination of the other
column vectors, or explain why this is impossible.
1D. Express v2as a linear combination of the other column vectors, or explain why this is impossible.
1E. Let b=7v1+6v2−12v4in the following two questions:
1E (i). Express bas a linear combination of the column vectors of Bwithout using v4(by replacing v4
with a LC of the other column vectors).
1E (ii). Can you express bas a linear combination of the column vectors of Bwithout using v2? Explain
your answer.
1 Challenge Bonus Question: Let mbe the number of rows of B. Suppose Bx=cdoes not have a
solution for every cin Rm. What is the RREF of B, where mis as small as possible?