Math 2360
Section I
1. (6 pts)
Final Exam
Form A
8/8/8
For each of the problems in the this section at least one of the choices is correct. For some of the
problems more than one of the choices is correct. Record your answers for problems in this
section on the answer sheet (last page).
⎡ 1 −2 0 3 ⎤
⎢
⎥
Consider the following matrix A = −2 1 3 3 . Which of the following matrices are
⎢
⎥
⎢⎣ −1 2 1 2 ⎥⎦
row-equivalent to A.
a.
d.
2. (8 pts)
⎡1
⎢ −2
⎢
⎢⎣ 0
⎡1
⎢0
⎢
⎢⎣ 0
−2 0 3 ⎤
1 3 3 ⎥⎥
0 1 2 ⎥⎦
0 0 3⎤
1 0 1 ⎥⎥
0 1 2 ⎥⎦
⎡1
⎢
b. 0
⎢
⎢⎣ 0
⎡1
⎢0
e.
⎢
⎢⎣ −1
−2 0 3 ⎤
1 3 3 ⎥⎥
0 1 2 ⎥⎦
−2 0 3 ⎤
−3 3 9 ⎥⎥
2 1 2 ⎥⎦
⎡1
⎢
c. 0
⎢
⎢⎣ 0
⎡1
⎢0
f.
⎢
⎢⎣ 0
0 0 7⎤
1 0 2 ⎥⎥
0 1 5 ⎥⎦
0 0 2⎤
1 −1 −3⎥⎥
0 0 0 ⎥⎦
Consider the following four matrices:
⎡ −3 0 1 ⎤
A=⎢
⎥
⎣ 1 2 −1⎦
⎡ 2 7 −1⎤
B=⎢
⎥
⎣ 3 −4 0 ⎦
⎡ −1 0 ⎤
D = ⎢⎢ 2 −3⎥⎥
⎢⎣ 1 1 ⎥⎦
⎡ 0 1⎤
C=⎢
⎥
⎣ −2 1⎦
Which of the following arithmetic operations are possible?
3. (8 pts)
a. 3A - 2B
b. AB
e. C 2
f.
( DA) 2
2
c.
AC - D
d. B
g.
D 2 A2
h. DB - BD
3
Consider the following subsets of \ :
a.
{( x1 , x2 , x3 )T | x1 > 0 or x2 > 0}
b.
{( x1 , x2 , x3 )T | ( x1 = 0) or ( x2 = 0) or ( x3 = 0)} .
c.
{( x1 , x2 , x3 )T | x1 − x3 = 0}
d.
{( x1 , x2 , x3 )T | x12 + x2 2 + x3 2 = 1}
Which of the above subsets of \ are subspaces of \ ?
3
3
4. (8 pts)
Consider the following subsets of P4 :
a.
{ p ∈ P4 | p (1) = 0}
b.
{ p ∈ P4 | p (1) ≥ p (0)}
c.
{ p ∈ P4 | p is even}
d.
{ p ∈ P4 | deg( p ) = odd }
Which of the above subsets of P4 are subspaces of P4 ?
5. (8 pts)
Consider the following subsets of \ :
3
a.
⎧ ⎡ 2 ⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 2 ⎥ ⎬
⎪⎢ 0 ⎥ ⎢ 3 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
c.
⎧ ⎡ −1⎤ ⎡ 5 ⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 3 ⎥ , ⎢ −6 ⎥ ⎬
⎪ ⎢ 0 ⎥ ⎢ −2 ⎥ ⎢ 4 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
b.
⎧⎡0⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 2 ⎥ , ⎢ −2 ⎥ , ⎢ 5 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 0 ⎥⎦ ⎢⎣ −2 ⎥⎦ ⎢⎣ −3⎥⎦ ⎭
d.
⎧⎡ 1 ⎤ ⎡ 3 ⎤ ⎡ 1 ⎤ ⎡ 0 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ , ⎢ −1⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 0 ⎥⎦ ⎢⎣ −1⎥⎦ ⎢⎣ 1 ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎭
Which of the above subsets of \ are spanning sets for \ ?
3
6. (8 pts)
3
Consider the following subsets of \ :
3
a.
⎧ ⎡1 ⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢1 ⎥ , ⎢ −1⎥ ⎬
⎪ ⎢0 ⎥ ⎢ 2 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
c.
⎧ ⎡ 0 ⎤ ⎡ −2 ⎤ ⎡ 1 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 0 ⎥⎦ ⎢⎣ −2 ⎥⎦ ⎢⎣ 3 ⎥⎦ ⎭
b.
⎧ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡ −6 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ −2 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣1 ⎥⎦ ⎢⎣ −2 ⎥⎦ ⎢⎣ 4 ⎥⎦ ⎭
d.
⎧ ⎡ 1 ⎤ ⎡ −2 ⎤ ⎡ − 2 ⎤ ⎡ 3 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ 2 ⎥ , ⎢1 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ −1⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎢⎣1 ⎥⎦ ⎭
Which of the above subsets of \ are linearly independent?
3
7. (4 pts)
Consider the set S = {1 + x ,1 − x ,1 − 2 x + x 2 } which is a subset of P3 . Is the set S linear
independent?
8. (8 pts)
3
Consider the following subsets of \ :
a.
⎧ ⎡ −1⎤ ⎡ −1⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −1⎥ , ⎢ −1⎥ ⎬
⎪⎢ 0 ⎥ ⎢ 1 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
c.
⎧ ⎡ 1 ⎤ ⎡ −1⎤ ⎡ −3 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ −1⎥ , ⎢ 2 ⎥ ⎬
⎪ ⎢ −1⎥ ⎢ 3 ⎥ ⎢ −6 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
b.
⎧ ⎡ 0 ⎤ ⎡ −2 ⎤ ⎡ −5 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ −2 ⎥ , ⎢ −2 ⎥ , ⎢ 1 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 3 ⎥⎦ ⎢⎣ 2 ⎥⎦ ⎢⎣ −4 ⎥⎦ ⎭
d.
⎧ ⎡ −1⎤ ⎡ −1⎤ ⎡ −1⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
⎨ ⎢ 0 ⎥ , ⎢ 1 ⎥ , ⎢ −3 ⎥ , ⎢ 0 ⎥ ⎬
⎪
⎪
⎩ ⎢⎣ 1 ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎭
Which of the above subsets of \ forms a basis for \ ?
3
9. (9 pts)
3
Determine whether the following are linear transformations from \ 4 to \ 3
⎡ x1 − 4 x2 + 4 x4 ⎤
⎥
x1 x2
a. L ( x ) = ⎢⎢
⎥
⎢⎣ −2 x1 + 2 x2 + x3 ⎥⎦
b.
⎡ x4 − 1/ x2 ⎤
⎥
L( x ) = ⎢⎢
0
⎥
⎢⎣ x4 + 1/ x ⎥⎦
x1
⎡
⎤
⎢
⎥
c. L( x ) = ⎢ x1 − x2 ⎥
⎢⎣ x1 − x2 + x3 ⎥⎦
10. (9 pts)
Determine whether the following are linear transformations from P3 to P4
x
a. L ( p ( x )) = x 2 p ′( x )
c.
11. (6 pts)
b.
L ( p ( x )) =
∫
p (t ) dt
0
L ( p ( x )) = x (1 + p ( x ))
Consider the sets of vectors given below. Which sets are orthogonal sets?
⎧ ⎡ − 1⎤ ⎡ 2 ⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎪
a. ⎨ 2 , −2 , 1 ⎬
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎪ ⎢ −2 ⎥ ⎢ − 3 ⎥ ⎢ 0 ⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎭
c.
{[1,
b.
⎧ ⎡ −1⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎪⎪ ⎢ 2 ⎥ ⎢ −1⎥ ⎪⎪
⎨ ⎢ −3 ⎥ , ⎢ −3 ⎥ ⎬
⎪⎢ 0 ⎥ ⎢ 2 ⎥ ⎪
⎪⎢ ⎥ ⎢ ⎥ ⎪
⎪⎩ ⎢⎣ 1 ⎥⎦ ⎢⎣ −5⎥⎦ ⎭⎪
−2, −1, 0, 3, −1] , [12, 3, 4, −4, −2, −4 ]
T
T
}
12. (8pts)
Which of the following sets forms an orthonormal basis for \ ?
2
1
⎧ 1
T
T ⎫
a. ⎨
[1, −2] , [ 2,1] ⎬
5
⎩ 5
⎭
c.
13. (9 pts)
{
T
⎡ 1 / 2, −1/ 2 ⎤ , ⎡1/ 2,1/ 2 ⎤
⎣
⎦ ⎣
⎦
T
}
b.
{[3 / 5, −4 / 5] , [3 / 5, 4 / 5] }
d.
{[−1, 0] , [0, −1] }
T
T
T
T
Let A be a 4 × 4 matrix. Which the following are (always) true?
a.
dim( N ( A)) = dim( R ( AT ))
b.
dim( N ( A)) = dim( R ( A))
c. dim( R ( A)) = dim( R ( AT ))
d.
N ( A) ⊥ R ( AT )
e.
N ( A) ⊥ R ( A )
f.
R ( A) ⊥ R ( AT )
g.
\ 4 = N ( A) ⊕ R ( AT )
h.
\ 4 = N ( A) ⊕ R ( A)
i.
\ 4 = R ( A) ⊕ R ( AT )
Section II
Answer the problems in this section on separate paper. You do not need to rewrite the problem
statements on your answer sheets. Work carefully. Do your own work.
Show all relevant supporting steps!
14. (9 pts)
i.
ii.
iii.
a.
15. (8 pts)
Each of the following augmented matrices is in row echelon form.
For each matrix, indicate whether the corresponding system of linear equations is consistent or
inconsistent
For each case in which the corresponding system of linear equations is consistent, indicate whether
the system has a unique solution or infinitely many solutions.
For each case in which the corresponding system of linear equations is consistent and has a unique
solution, find that unique solution.
⎡ 1 −1
⎢
⎢0 1
⎢0 0
⎢
⎢⎣ 0 0
1 0⎤
⎥
0 0 2⎥
1 −1 1 ⎥
⎥
0 1 0 ⎥⎦
0
⎡ 1 −1
⎢
0 1
b. ⎢
⎢0 0
⎢
⎢⎣ 0 0
1 0⎤
⎥
0 1⎥
1 −1 −1⎥
⎥
0 0 1 ⎥⎦
0
0
⎡ 1 −1
⎢
0 1
⎢
c.
⎢0 0
⎢
⎢⎣ 0 0
1 0⎤
⎥
0 1 −1⎥
1 −1 −2 ⎥
⎥
0 0 0 ⎥⎦
0
The following augmented matrix is in reduced row echelon form. Find the solution set of the
corresponding system of linear equations.
⎡1
⎢
0
a. ⎢
⎢0
⎢
⎣⎢ 0
0 2 0 1 2⎤
⎥
1 −1 0 2 1 ⎥
0 0 1 −2 −1⎥
⎥
0 0 0 0 0 ⎦⎥
16. (9 pts)
17. (9 pts)
For each of the following pairs of matrices find an elementary matrix E such that EA =B.
⎡ −1 0 2 ⎤
⎢
⎥
a. A = 1 −1 −1
⎢
⎥
⎢⎣ 2 −1 2 ⎦⎥
⎡ −1 0 2 ⎤
B = ⎢⎢ 2 −1 2 ⎥⎥
⎢⎣ 1 −1 −1⎥⎦
⎡ −1 2 0 −2 ⎤
⎢ 2 2 −1 3 ⎥
⎥
b. A = ⎢
⎢ 1 2 −1 4 ⎥
⎢
⎥
⎣ 0 −1 2 −3 ⎦
⎡ −1 2 0 −2 ⎤
⎢ 2 0 3 −3 ⎥
⎥
B=⎢
⎢ 1 2 −1 4 ⎥
⎢
⎥
⎣ 0 −1 2 −3 ⎦
⎡ 1 − 2 0 2 −2
⎢ −2 1 −1 −1 −3
⎢
A
=
Consider the matrix A given by
⎢ −1 2 −1 0 2
⎢
⎣ −2 −2 −3 4 −9
−1⎤
2 ⎥⎥
. A straightforward
−1⎥
⎥
0⎦
reduction by elimination shows that A is row equivalent to U where
⎡1
⎢0
U =⎢
⎢0
⎢
⎣0
0
1
0
0
0 4 / 3 0 −7 / 3 ⎤
0 −1/ 3 0 −2 / 3⎥⎥
.
1 −2 0 2 ⎥
⎥
0
0 1 0 ⎦
a. Find a basis for the row space of A.
b. Find a basis for the column space of A.
c. Find a basis for the null space of A.
18. (8 pts)
Consider the linear transformation mapping \ to \ given by
3
3
L ( x ) = ( x1 − 2 x2 − 3 x3 , x1 − x2 , x1 + 3 x3 )T
a. Find the kernel of L.
b. Find the dimension of the range of L.
19. (8 pts)
Consider the linear transformation mapping \ to \ given by
3
2
L ( x ) = ( x1 − 2 x2 + x3 , x2 + 2 x3 )T
Find a matrix A which represents L with respect the standard basis [ e1 , e 2 , e3 ] in \ and the
3
⎡1⎤
⎡ −1⎤
2
ordered basis [ b1 , b2 ] in \ where b1 = ⎢ ⎥ and b2 = ⎢ ⎥ .
⎣ −1⎦
⎣0⎦
20. (8 pts)
Find the equation of the plane in \ which is normal to the vector N = [1, − 1, 2]T and which
3
passes through the point P0 = [ −1, 2, − 5]T .
21. (8 pts)
Find the distance in \ from the point P0 = [ −1, − 3, 1]T to the plane given by
3
2 x − 3 y + z = −4 .
22. (8 pts)
Let S be the subspace in \ which is spanned by the set
3
{[1, − 2, 1]
T
, [ −1, 2, 2 ]
T
} . Find
a basis for S ⊥ .
23. (8 pts)
Consider the following orthonormal basis U = [ u1 , u2 , u3 ] for \ , where
3
u1 =
1
1
1
T
T
T
[1, − 1, 1] , u2 = [ −1, 0, 1] and u3 = [1, 2, 1] .
3
2
6
Suppose L is a linear transformation from \ to \ which satisfies L ( u1 ) = [ −1, 0, 2, 1]T ,
3
4
L ( u2 ) = [2, − 3, 0, − 1]T and L ( u3 ) = [1, 1, − 1, − 3]T . Find the value of
L ([ −1, 2, 1]T ) .
24. (8 pts)
Let S be the subspace of \ spanned by the vectors u1 and u2 (given in Problem 23). Let
3
x = [1, 3, − 2]T . Find the projection of x onto S.
25. (10 pts)
Let E be the set of linearly independent vectors E = [ x1 , x 2 , x3 ] , where
x1 = [1, − 1, 0, − 1] , x 2 = [1, 0, 0, − 1] and x3 = [1, − 2, − 1, 0 ] .
T
T
T
Find (using the Gram-Schmidt orthogonalization process) a set U of orthonormal vectors so that
span ( E ) = span (U ) .
26. (8 pts)
⎡3 1 0⎤
Let A = ⎢ −6 −2 0 ⎥ . Find the eigenvalues of A.
⎢
⎥
⎢⎣ 4
2 −1⎥⎦
27. (8 pts)
⎡ 1 −2 − 2 ⎤
Let A = ⎢0 0 −1⎥ . Determine whether A is diagonalizable.
⎢
⎥
⎢⎣0 0 1 ⎥⎦
Name _________________________
Form A
Answers
1.
a. Yes
No
b. Yes
No
c. Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
2.
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
e. Yes
No
f. Yes
No
g. Yes
No
h.
Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
a. Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d.
Yes
No
a. Yes
No
b. Yes
No
c. Yes
No
d. Yes
No
e. Yes
No
f. Yes
No
g. Yes
No
h. Yes
No
i. Yes
No
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.