Math 205
Homework 5
Due Tuesday October 3rd at 11:59pm
Sections 4.4-4.6
For the first three questions, consider the following vectors in R3 and a 3 × 3 matrix:
 
 
 


 


1
3
2
1
1
1 3 2
v 1 =  2  , v2 =  8  , v3 =  6  , y =  6  , z =  6  , A =  2 8 6 
1
2
1
−1
1
1 2 1
1. (a) Determine whether or not y is in span{v1 , v2 , v3 }.


x1
(b) Find the solution set to the matrix equation Ax = y, where x =  x2  .
x3
Hint: You may use the results from (a) to explain your answer.
2. (a) Determine whether or not z is in span{v1 , v2 , v3 }.


x1
(b) Find the solution set to the matrix equation Ax = z, where x =  x2  .
x3
Hint: You may use the results from (a) to explain your answer.
3. Determine whether or not {v1 , v2 , v3 } span R3 . If not, find a vector in R3 that is not
in the span of the vectors.
4. Determine whether the set P in P2 (R) is linearly independent or not. If not, determine
a linearly independent set that spans same subspace of P2 (R) as that spanned by P and
find the relation of linear dependence among the vectors in P .
P = {2 + x2 , 4 − 2x + 3x2 , 1 + x}
5. Determine whether the following sets of vectors are linearly independent or not.
1
(a) S1 = {1 − 2x + 4x2 , 3 − 6x + 12x2 } in P2 (R)
(b) S2 = {2x, 1 + x, 1 + x + x2 , 2x + x3 , 2 − x + x2 + 2x3 } in P3 (R)
(c)

 
1
2



  −1
1
 
S3 = 
 −1  ,  3



1
1

1 

  1 
, 
  2 


1
 
in R4
(d) S4 = {1, cos 3x, sin 3x} in C 2 (R).
6. Given a subset M and a set of vectors in M2 (R)
−2 1
0 2
3 1
T
M = {A ∈ M2 (R)|A = A }, V =
,
,
1 4
2 1
1 0
(a) Show that M is a subspace of M2 (R).
(b) Show that V is a basis of M .
7. Determine whether the following sets of vectors are bases for R4 .
(a)
  
1


  
1  
V1 = 
 0 ,



2
 
2
−1


1   1
,
3   1
−1
−2








(b)
  
1
2


  
1   1
V2 = 
,

0   3



2
−1
 
 
−1
2
  1   −1
,
 
  1 , 1
−2
2








8. Determine whether the following sets of vectors are bases for M2 (R).
(a)
S1 =
−2 −8
1
4
0 1
−5 0
3 −2
,
,
,
−1 1
5 −4
4 −1
(b)
S2 =
−2 −8
1
4
0 1
−5 0
3 −2
1 2
,
,
,
,
−1 1
5 −4
4 −1
3 4
9. Given a set P of P2 (R)
P = {1 + x, −x + x2 , 1 + 2x2 }.
(a) Use the Wronskian to show that P is a basis for P2 (R).
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(b) Find the polynomial q = 1 − 2x − 3x2 as a linear combination of P .
10. Find the basis of the null space of the given matrix A.


1 1 −1 1
A =  2 −3 5 −6 
5 0
2 −3
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