Math 2360
Exam II
Form A - Make Up
23 July 2008
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). 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 following four systems of linear equations:
i.
= −3
⎧ x1 − x2
⎪
⎨ 2 x1 − x 2 + 3 x3 = −2
⎪ − x + 2 x + 3x = 1
2
3
⎩ 1
iii.
⎧ 2 x1 − x2 + x3 = −1
⎪
− x 2 + x3 = 3
⎨
⎪ 3x − x − x = 0
2
3
⎩ 1
ii.
⎧ x1 − x2 + x3 = 0
⎪
⎨ x1 − x2 + 3 x3 = −1
⎪
⎩ − x1 + 2 x2 − 2 x3 = −2
iv.
⎧ 2 x1 − x2 + x3 = 0
⎪
⎨ x1 − x2 + 2 x3 = −1
⎪ 3x − x
= 2
2
⎩ 1
Which of the above fours systems of linear equations can be solved by Cramer’s Rule?
2. (12 pts)
Consider the set V = {( x1 , x2 )T | x1 , x2 ∈ \} . Define “addition” by
( x1 , x2 )T ⊕ ( y1 , y2 )T = ( x1 + y1 , x2 − y2 )T . (We will use symbol ⊕ for “addition” since this is not
the usual addition for order pairs.) Define scalar multiplication by
α ( x1 , x2 ) = (α x1 , α x2 ) .
(This is usual scalar multiplication for ordered pairs.)
Recall the vector space axioms for addition for a vector space V.
i.
Addition is commutative, i.e. for every x, y in V we have x ⊕ y = y ⊕ x
ii.
iii.
Addition is associative, i.e., for every x, y, z in V we have ( x ⊕ y ) ⊕ z = x ⊕ ( y ⊕ z )
Existence of an additive identity in V, i.e. there exists an additive identity (called 0) in V so that
for every x in V we have x ⊕ 0 = x
Existence of additive inverses in V, i.e. for each x in V there exists an additive inverse (called -x)
in V so that x ⊕ − x = 0
iv.
Which of the above axioms are not satisfied by V with its defined “addition” and scalar multiplication?
3. (9 pts)
Consider the following subsets of \ :
2
i.
{( x1 , x2 )T | 2 x1 + 3 x2 = 1}
iii.
{( x1 , x2 )T | x12 = x2 2 }
ii.
{( x1 , x2 )T | 2 x1 − x2 = 0}
Which of the above subsets of \ are subspaces of \ ?
2
2
4. (9 pts)
Consider the following subsets of P3 :
i.
{ p ∈ P3 | p ( −1) + p (1) = 0}
iii.
{ p ∈ P3 | p ′(1) = 0}
ii.
{ p ∈ P3 | p (1) > 0}
Which of the above subsets of P3 are subspaces of P3 ?
5. (12 pts)
Consider the following subsets of \ :
3
i.
⎧ ⎡ 1 ⎤ ⎡ 0 ⎤ ⎡ −2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢1 ⎥ , ⎢ 3 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢1 ⎥ ⎢ −1⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
iii.
⎧ ⎡ 1 ⎤ ⎡ −1⎤ ⎡ 0 ⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ 1 ⎥ , ⎢ 2 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢ −1⎥ ⎢ 4 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
ii.
⎧ ⎡ −1⎤ ⎡ −1⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 1 ⎥ , ⎢ 3 ⎥ , ⎢ −2 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ −1⎥⎦ ⎢⎣ 2 ⎥⎦ ⎢⎣ 2 ⎥⎦ ⎭
iv.
⎧ ⎡ 2 ⎤ ⎡ −1⎤ ⎡ 1 ⎤ ⎡ 0 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 1 ⎥ , ⎢ −1⎥ , ⎢ 2 ⎥ , ⎢ −1⎥ ⎬
⎪ ⎢ 3 ⎥ ⎢ −1⎥ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
Which of the above subsets of \ are spanning sets for \ ?
3
6. (12 pts)
3
Consider the following subsets of \ :
3
i.
⎧ ⎡ −2 ⎤ ⎡ 4 ⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ − 1⎥ , ⎢ 2 ⎥ , ⎢ 3 ⎥ ⎬
⎪ ⎢ 1 ⎥ ⎢ −2 ⎥ ⎢ 2 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
iii.
⎧ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ −3⎥ ⎬
⎪ ⎢ −1⎥ ⎢ −2 ⎥ ⎢ 3 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
ii.
⎧ ⎡ 1 ⎤ ⎡ − 2 ⎤ ⎡ − 1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ ⎬
⎪ ⎢ 1 ⎥ ⎢ −2 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
iv.
⎧ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 2 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 3 ⎥⎦ ⎢⎣ −6 ⎥⎦ ⎭
Which of the above subsets of \ are linearly independent?
3
7. (5 pts)
Consider the set S = {1 + x,1 − x + x 2 , 2 − 2 x − x 2 } which is a subset of P3 . Is the set S linear
independent?
8. (5 pts)
Consider the set S = {cos 2π x, sin 2 π x, cos 2 π x} which is a subset of C 2 ( −∞ , ∞ ) . Is the set S
linear independent?
9. (9 pts)
Consider the following subsets of \ :
2
i.
⎧ ⎡ 2 ⎤ ⎡ − 1⎤ ⎫
⎨⎢ ⎥ , ⎢ ⎥ ⎬
⎩ ⎣ 1 ⎦ ⎣ −2 ⎦ ⎭
iii.
⎧ ⎡ −2 ⎤ ⎡ 3 ⎤ ⎫
⎨⎢ ⎥ , ⎢ ⎥ ⎬
⎩ ⎣ −4 ⎦ ⎣ −6 ⎦ ⎭
ii.
⎧ ⎡ −1⎤ ⎡ 3 ⎤ ⎫
⎨⎢ ⎥ , ⎢ ⎥ ⎬
⎩ ⎣ 2 ⎦ ⎣ −6 ⎦ ⎭
Which of the above subsets of \ form a basis for \ ?
2
10. (12 pts)
2
3
Consider the following subsets of \ :
i.
⎧ ⎡ 1 ⎤ ⎡ − 1⎤ ⎡ 3 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ −2 ⎥ , ⎢1⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢ − 2 ⎥ ⎢1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
iii.
⎧ ⎡1 ⎤ ⎡ −1⎤ ⎡ −3⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ ⎬
⎪ ⎢ 1 ⎥ ⎢ − 3⎥ ⎢ 6 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
ii.
⎧⎡ 1 ⎤ ⎡2⎤ ⎡ 3 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 0 ⎥ , ⎢ −1⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 0 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎭
iv.
⎧ ⎡ 1 ⎤ ⎡ −1⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ −3⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 2 ⎥⎦ ⎢⎣ −2 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎭
Which of the above subsets of \ forms a basis for \ ?
3
11. (4 pts)
3
Consider the following statement:
Theorem. Let {a1 , a 2 , a3 ," , a n } ⊂ \ n and let A = [ a1 a 2 a3 " a n ] . The following are equivalent:
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
tr( A) ≠ 0
vii.
{a1 , a2 , a3 ," , an } is a basis for \ n
viii.
ix.
x.
dim( span ({a1 , a 2 , a3 ," , a n })) = n
N ( A) = {0}
A≠0
Which two of the above statements need to be deleted to make the theorem true?
Name _________________________
Form A
Answers
1.
i. Yes
No
ii. Yes
No
iii. Yes
No
iv. Yes
No
2.
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
i. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
i. Yes
No
ii. Yes
No
iii. Yes
No
iv. Yes
No
3.
4.
5.
6.
7.
8.
9.
10.
11. ______, ______