Download (3s)

name: ραλφ
Mathematics 220 third homework
due Thursday, July 31, 2014
5. For each of the matrices A and C below, find implicit and explicit descriptions of the column space of the matrix and of the null space of the matrix.
Row reduction information is given so that you shouldn’t need to do many
calculations.


2 1 6
 1 3 −2 

A=
 3 1 10 
4 1 14




2 1 6 b1
1 0 4
−b3 + b4
 1 3 −2 b2 
 0 1 −2

4b3 − 3b4




 3 1 10 b3  reduces to  0 0 0
b1 − 2b3 + b4 
4 1 14 b4
0 0 0 b2 − 11b3 + 8b4



b1





b1 − 2b3 + b4 = 0 
b2 
4 

(implicit) col A = 
∈R b3 
b2 − 11b3 + 8b4 = 0 





b4


 
1 
2




 3 
1 




,
(explicit) col A = Span 
3   1 





1
4




 x1
x + 4x3 = 0
(implicit) null A =  x2  ∈ R3 1
x2 − 2x3 = 0 

x3



 

−4x3 
 x1
 −4 
(explicit) null A =  x2  =  2x3  = Span  2 




x3
x3
1

2
C= 3
4

2 1 2 5
 3 2 5 8
4 1 0 9

b1
b2 
b3
reduces to
1
2
1
2
5
0

5
8 
9

1
0

 0 1

0 0
−1
2
− 51 b2 + 25 b3

4
1
0
0
4
3
5 b2 − 5 b3
b1 − 52 b2 − 15 b3





 b1
(implicit) col A =  b2  ∈ R3 b1 − 52 b2 − 15 b3 = 0

b3


 
1 
 2
(explicit) col A = Span  3  ,  2 


1
4


x1



x2 
 ∈ R4 x1 − x3 + 2x4 = 0
(implicit) null A = 
x2 + 4x3 + x4 = 0


x3



x4

 
x1
x3 − 2x4



 −4x3 − x4
x2 



(explicit) null A = 
=
x3  
x3



x4
x4











 

1






 


 = Span  −4  , 
 


1





0

−2 


−1 

0 


1
6. Explain why each of the following subsets of Rn is or is not a vector space:
(a)
(x, y) ∈ R2
| x2 − y 2 = 0
not a vector space, because it is not closed under addition:
x
1
1
1
1
2
=
and
are in the set, but
+
=
is not.
y
1
−1
1
−1
0
(b)





 b1
 x1 + 2x2 = b1 

 b2  ∈ R3 such that
4x1 + 3x2 = b2
has a solution x1 and x2




b3
5x1 − x2 = b3

1
a vector space, since this is equal to the column space of the coefficient matrix  4
5

2
3 
−1
(c)





 b1
 x1 + 2x2 = b1 

 b2  ∈ R3 such that
4x1 + 3x2 = b2
does not have a solution x1 and x2




b3
5x1 − x2 = b3
not a vector space, since the zero vector in R3 is not in the set.




 x1 + 2x2 = 3  
x1
4x1 + 3x2 = 7
(d)
∈ R2 such that
 x2


5x1 − x2 = 4
not a vector space, since the zero vector in R2 is not in the set.
continuation of problem (6):
(e)



x1
x2


 x1 + 2x2 = 0  
4x1 + 3x2 = 0
∈ R2 such that


5x1 − x2 = 0

1 2
a vector space, since this is equal to the null space of the coefficient matrix  4 3 
5 −1

7. Explain why each of the following sets of functions is or is not a vector space:
R1
(a) the set of continuous functions with domain [0, 1] such that 0 f (x) dx = 0.
this is a vector space:
it’s closed under scalar multiplication:
R1
If a function f is in the set, then 0 f (x) dx = 0.
R1
R1
Is cf in the set ? Check the condition: 0 cf (x) dx = c 0 f (x) dx = c · 0 = 0.
it’s closed under addition:
R1
R1
If a functions f and g are the set, then 0 f (x) dx = 0 and 0 g(x) dx = 0
Is f + g in the set ? Check the condition:
Z 1
Z 1
Z 1
f (x) + g(x) dx =
f (x) dx +
g(x) dx = 0 + 0 = 0.
0
0
0
(b) the set of continuous functions with domain [0, 1] which have inverse functions.
This is not a vector space. The zero function f (x) = 0 is not in the set since it
is not one-to-one and has no inverse function.
(c) the set of differentiable functions with domain R which satisfy the differential
equation f 00 (x) + 9f (x) = 0.
this is a vector space:
it’s closed under scalar multiplication:
If a function f is in the set, then f 00 (x) + 9f (x) = 0.
Is cf in the set ? Check the condition: (cf )00 (x)+9(cf (x)) = cf 00 (x)+9cf (x) =
c(f 00 (x) + 9f (x)) = 0.
it’s closed under addition:
If a functions f and g are the set, then f 00 (x)+9f (x) = 0 and g 00 (x)+9g(x) = 0
Is f + g in the set ? Check the condition:
(f +g)00 (x)+9(f (x)+g(x)) = f 00 (x)+g 00 (x)+9f (x)+9g(x) = f 00 (x)+9f (x)+g 00 (x)+9g(x) = 0+0 = 0
8. Determine whether the set of all 2 × 2 matrices which satisfy the equation
AAT = I is or is not a vector space. Explain.
This is not a vector space since the 2 × 2 zero matrix is not in the set, since
00T 6= I.
9. Let C 2 be the set of functions with domain R which have first and second derivatives at all points. Define a linear transformation T : C 2 → C 2
by T (f ) = f 00 , in other words, each input function is mapped to its second
derivative function.
What is the kernel of this transformation?
00
f | f (x) = 0} =
Z Z
f | f (x) =
Z
0 dx =
c dx = cx + c2
The kernel of T is the set of linear functions.
10. Let M2×2 be the set of two by two matrices with real entries, and define a
linear transformation T : M2×2 → M2×2 by
T (A) = A − AT
Describe the kernel of this transformation.
A A − AT = 0 = A | A = AT
The kernel of T is the set of symmetric matrices, which have the format
a b
b c