Group Solve 2
Math 220
Names: _________________________________________
Write all explanations in complete, correct English sentences.
1. Consider a homogeneous system of seven linear equations in nine
variables. Suppose that there are two solutions of the system that are
not multiples of one another, and all other solutions are linear
combinations of these solutions. Will the associated nonhomogeneous system have a solution for every possible choice of
constants on the right sides of the equations? Justify your answer.
Mention any theorems that you use.
2 Suppose   b1,b2 
3
 4
b

where 1 5 , b 2  6 , and E  e1,e2  where
 
 
1
0 
e1    , e 2    .
0 
1
Find the change of coordinates matrix to go from β to the standard
basis and from the standard basis to β. Label your answers clearly.
No explanations are needed if work is clearly shown.
3. Assume A  a1 a2 a3 a4 a5 
and
row equivalent, where
2 2
0
7
 1
1
 2  3
0
1  1  5

A
B
 3  4
0
0  2  3 and



6 6
5
1
 3
0
a. Find a basis for the column space
needed.
B  b1 b2 b3 b4 b5  are
0
4 0  3
1 3 0
5
0
0 1  4

0
0 0
0
of A. No explanations are
b. Find a basis for Row A. No explanations are needed.
c. Fill in the blank. Nul A is a _____ dimensional subspace of R__. No
explanations are needed.
4. Determine whether the following sets are subspaces. Justify your
answer. Mention any theorems that you use.
a 



G  b  : a  0, b  0, c  0
a.
 c 

 

 a  3b 




 a  b 

H 
: a, b  R 
b.
  2a  b 



 4a 

5. Prove the Theorem: If A is an m x n matrix, then Nul A is a
subspace of Rn.
6. a. Complete the statement of the Unique Representation Theorem.
Let β = {v1, v2, … , vn} be a basis for a vector space V. Then for
each x in V,
b. Explain why a linearly independent set of four vectors in P3 is a
basis for P3. Mention the appropriate fact(s) from Chapter 4.