Math 205B Quiz 03 page 1 10/02/2009 Name
1. Let Sbe a set {v1,v2, . . . , vp}of vectors in Rm. Define what it means to say the set Sis linearly independent.
2. Let A=
1 6 3 3 1
1 7 3 3 1
2 13 7 9 4
1 6 5 9 5
and label its column vectors as c1,c2, . . . , c5. It’s a fact that the RREF of
1 6 3 3 1 1 0 0 0
1 7 3 3 1 0 1 0 0
2 13 7 9 4 0 0 1 0
1 6 5 9 5 0 0 0 1
is
100−6−5 0 −23 17 −10
0 1 0 0 0 0 3 −2 1
0 0 1 3 2 0 1 −1 1
0 0 0 0 0 1 2 −2 1
.
2A. Let b=
b1
b2
b3
b4
. Use the above information to decide what conditions, if any, b1,b2,b3, and b4must satisfy in order
for Ax=bto have a solution.
2B. Without doing any work, find an easy solution of Ax=c3.
2C. Since Ax=c3does have a solution, the entries of c3must satisfy the conditions in (2A). Show that this is so.
3. Let Aand c1,c2, . . . , c5be as in problem (2).
3A. Is the set S={c1,c2, . . . , c5}linearly independent? Explain.
3B. Show that it is possible to write c3as a linear combination of the other four vectors. (Give an explicit LC).
3C. Explain why it is impossible to write c2as a linear combination of the other four vectors.