Math 2360
Exam II
Make-up Form A
28 October 2009
Section I
For each of the following problems at least one of the choices is correct. For many of the problems more than
one of the choices is correct. Record your answers for each problem on the answer sheet (last page). For theis
section, turn in (only) the answer sheet (with your answers on it) at the end of the exam. Do your own work.
You may keep the exam for your own records.
1. (12 pts)
Consider the V set of polynomials with degree less than 3. Define “addition” for V by if
p , q ∈ V with p ( x ) = a + bx + cx 2 and q ( x ) = α + β x + γ x 2 , then p ⊕ q is defined by
p ⊕ q ( x ) = aα + b β x + cγ x 2 . (We will use symbol ⊕ for “addition” since this is not the usual
addition for polynomials.) Define scalar multiplication for V by if p ∈ V with
p ( x ) = a + bx + cx 2 and α ∈ \ then α p is defined by α p ( x ) = α a + α bx + α cx 2 . (This is
usual scalar multiplication for polynomials.)
Recall the vector space axioms for addition for a vector space V.
i.
ii.
iii.
iv.
Addition is commutative, i.e. for every p, q in V we have p ⊕ q = q ⊕ p
Addition is associative, i.e., for every p, q, r in V we have ( p ⊕ q ) ⊕ r = p ⊕ (q ⊕ r )
Existence of an additive identity in V, i.e. there exists an additive identity (called 0) in V so that
for every p in V we have p ⊕ 0 = p
Existence of additive inverses in V, i.e. for each p in V there exists an additive inverse (called -p)
in V so that p ⊕ − p = 0
Which of the above axioms are not satisfied by V with its defined “addition” and scalar multiplication?
2. (9 pts)
Consider the following subsets of \ 4 :
i.
{( x1 , x2 , x3 , x4 )T | x1 − 2 x2 = 3 x3 − 4 x4 }
iii.
{( x1 , x2 , x3 , x4 )T | x1 = 0 and x2 = 0 and x3 = 0 and x4 = 0}
{( x1 , x2 , x3 , x4 )T | x1 x2 = x3 + x4 } ii.
Which of the above subsets of \ 4 are subspaces of \ 4 ?
3. (9 pts)
Consider the following subsets of P4 :
i.
{ p ∈ P4 | p is an odd function and p (1) = 1}
iii.
{ p ∈ P4 | p ′′(2) = 0}
Which of the above subsets of P4 are subspaces of P4 ?
ii.
{ p ∈ P4 | p ( −1) = p (1)}
4. (12 pts)
Consider the following subsets of \ :
3
i.
⎧ ⎡ 1 ⎤ ⎡0 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −2 ⎥ , ⎢ 1 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
iii.
⎧ ⎡ 0 ⎤ ⎡ −1⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ 0 ⎥ ⎬
⎪ ⎢ 2 ⎥ ⎢ −1⎥ ⎢ −1⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
ii.
⎧ ⎡ −1⎤ ⎡ 1 ⎤ ⎡ 0 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 1 ⎥ , ⎢ 2 ⎥ , ⎢ −2 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 0 ⎥⎦ ⎢⎣ −2 ⎥⎦ ⎢⎣ 4 ⎥⎦ ⎭
iv.
⎧ ⎡ 1 ⎤ ⎡ −1⎤ ⎡ 0 ⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ , ⎢ −1⎥ ⎬
⎪ ⎢ 1 ⎥ ⎢ −1⎥ ⎢ −2 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
Which of the above subsets of \ are spanning sets for \ ?
3
5. (12 pts)
3
Consider the following subsets of \ :
3
i.
⎧ ⎡ −1⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −2 ⎥ , ⎢ 2 ⎥ ⎬
⎪⎢ 1 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
iii.
⎧ ⎡ −1⎤ ⎡ 1 ⎤ ⎡ 0 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 1 ⎥ , ⎢ 2 ⎥ , ⎢ −2 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢ −2 ⎥ ⎢ 4 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
ii.
⎧ ⎡ 0 ⎤ ⎡ −1⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ 0 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 2 ⎥⎦ ⎢⎣ −1⎥⎦ ⎢⎣ −1⎥⎦ ⎭
iv.
⎧ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡ −2 ⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ − 1⎥ , ⎢ 0 ⎥ ⎬
⎪ ⎢1 ⎥ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
Which of the above subsets of \ are linearly independent?
3
6. (6 pts)
i.
Consider the following subsets P3 .
S = {1 − x + x 2 ,1 − 2 x + x 2 ,1 + 3 x + x 2 }
Which of the above subsets of P3 are linearly independent?
ii. S = {1,1 + x ,1 + x + x 2 }
7. (9 pts)
Consider the following subsets of \ :
2
i.
⎧ ⎡ 2 ⎤ ⎡ − 1⎤ ⎫
⎨⎢ ⎥ , ⎢ ⎥ ⎬
⎩ ⎣ 1 ⎦ ⎣ −2 ⎦ ⎭
iii.
⎧ ⎡1 ⎤ ⎡ − 2 ⎤ ⎡ 3 ⎤ ⎫
⎨⎢ ⎥ , ⎢ ⎥ , ⎢ ⎥ ⎬
⎩ ⎣ 3 ⎦ ⎣ −4 ⎦ ⎣ −2 ⎦ ⎭
⎧ ⎡ −3 ⎤ ⎡ 2 ⎤ ⎫
⎨⎢ ⎥ , ⎢ ⎥ ⎬
⎩ ⎣12 ⎦ ⎣ −8 ⎦ ⎭
ii.
Which of the above subsets of \ form a basis for \ ?
2
8. (12 pts)
2
3
Consider the following subsets of \ :
i.
⎧⎡ 1 ⎤ ⎡2⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 0 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
iii.
⎧ ⎡1 ⎤ ⎡ −1⎤ ⎡ −2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ ⎬
⎪ ⎢ 1 ⎥ ⎢ −3⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
ii.
⎧ ⎡ 1 ⎤ ⎡ −1⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ −2 ⎥ , ⎢ 1 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢ −2 ⎥ ⎢ 2 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
iv.
⎧ ⎡1 ⎤ ⎡ −1⎤ ⎡ −1⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ −3 ⎥ , ⎢ 1 ⎥ ⎬
⎪ ⎢1 ⎥ ⎢ −1⎥ ⎢ 1 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
Which of the above subsets of \ forms a basis for \ ?
3
9. (3 pts)
3
Consider the following statement:
Theorem. Let {a1 , a 2 , a3 ," , a n } ⊂ \ and let A = [ a1 a 2 a3 " a n ] . The following are equivalent:
n
i.
ii.
iii.
A is invertible
Ax = 0 has a unique solution
A is row equivalent to I
iv.
v.
vi.
Ax = b has a solution for every b ∈ \ n
det( A) ≠ 0
diag( A) ≠ 0
vii.
{a1 , a2 , a3 ," , an } is a basis for \ n
viii.
ix.
dim( span ({a1 , a 2 , a3 ," , a n })) = n
N ( A) = {0}
Which one of the above statements needs to be deleted to make the theorem true?
10. (3 pts)
Let A be a 6 × 14 matrix. If the dimension of the null space of A is 9, then the rank of A is:
a.
e.
Section II
11. (15 pts)
3
b.
5
c.
8
Undeterminable from the given information
d.
9
Answer the problem in this section on back of the answer sheet. You do not need to rewrite the
problem statement. Work carefully. Do your own work. Show all relevant supporting steps!
⎡
⎢
Consider the matrix A given by A = ⎢
⎢
⎢
⎣
1 −2 −1
1⎤
3 −3 0 3
4 ⎥⎥
. A straightforward
−1 − 1 −2 1
−2 ⎥
⎥
1 −8 − 7 8 − 1⎦
2
reduction by elimination shows that A is row equivalent to U where
⎡
⎢
U =⎢
⎢
⎢
⎣
1
0
0
0
0 1 0 5 / 3⎤
1 1 −1 1/ 3 ⎥⎥
.
0 0 0
0 ⎥
⎥
0 0 0
0 ⎦
a.
b.
c.
Find a basis for the row space of A.
Find a basis for the column space of A.
Find a basis for the null space of A.
Answer Sheet - Form A
Name _________________________
1.
i. Is Not Satisfied
Is Satisfied
ii. Is Not Satisfied
Is Satisfied
iv. Is Not Satisfied
Is Satisfied
iii. Is Not Satisfied
Is Satisfied
i. Yes
No
ii. Yes
No
iii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
iv. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
iv. Yes
No
i. Yes
No
ii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
iv. Yes
No
2.
3.
4.
5.
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9.
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10.
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