MATH 218 Final, Summer 2016
American University of Beirut · Elementary Linear Algebra with Applications · 11 pages
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July 23, 2016. MATHEMATICS 218 NAME. -------------------
Time : 90 minutes Final Exam ID# . ------------------
Summer 2015-16
Circle your section number :
Rana Nassif
Hazar Abu-Khuzam
Wael Mahboub
Michella Bou Eid
3
4
2
6
1
5
8:45 am
10:00 am
12:30 pm
11:15 am
1:45 pm
7:30 am
PROBLEM GRADE
PART I
1
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2
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3 ------- / 7
4. ------- / 10
5. ------- / 6
PART II
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15
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6-15 ------- / 30
PART III Answer True or False only in the table below:
16
17
18
19
20
21
22
23
24
25
26
27
T
T
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
F
F
F
F
16-27 ------- / 24
TOTAL ------- / 100
1
PART I. Answer each of the following problems in the space provided for each
problem ( Problem 1 to Problem 5).
1. Let A=
(a) Find the eigenvalues and a basis for each eigenspace of A.
[ 8 points]
(b) Show that A is diagonalizable.
2
[2 points]
(c) Find an invertible matrix P and a diagonal matrix
D such that P-1AP=D. ( Do not verify)
[2 points]
2. Let
be the linear transformation defined by
(a) Find a general formula for
[3 points]
(b) Find a basis for N(T)
3
[4 points]
(c) Find a basis for the range R(T)
[4 points]
3. Let
be the linear operator defined by
4
(a) Find the matrix of T,
, where B={1,x,x2} is the standard ordered basis of P2,
and
is another ordered basis of P2.
[4 points]
(b) Find the transition matrix
[3 points]
4. Let W=
be the subspace of R3 with the usual dot product.
5
(a) Find a basis for
.
[5 points]
(b) Use pat (a) and the Gram-Schmidt process to find an orthogonal basis for
.
[5 points]
5. Let
be a linear operator on a vector space V such that dim N(T)=1 and
and
Show that
are linearly dependent in V.
[6 points]
6
PART II. Circle the correct answer for each of the following problems
( Problem 6 to Problem 15). IN THE TABLE IN THE FRONT PAGE
[3 points for each correct answer]
6. Let W=
. Then dim (W)=
a.
0
b.
1
c.
2
d.
3
e.
None of the above
[3 points]
7. Let V = P2 with inner product given by
7
Let W be the subspace of V with basis = {1, x}. Then
a.
{1, x} is an orthogonal basis for W
b. {1, ½ x} is an orthogonal basis for W
c.
{1, 2x} is an orthogonal basis for W
d. {1, 3x} is an orthogonal basis for W
e.
None of the above
[3 points]
8.
Let A be an nxn matrix such that An=0. Which one of the following is False:
a.
λ=0 is an eigenvalue for A
b.
λ=0 is an eigenvalue for AAt
c.
=0
d.
A=0
None of the above
[3 points]
9.
Consider the matrix
A=
, then
a.
A has only 2 linearly independent eigenvectors.
b.
A is invertible.
c.
A is diagonalizable.
d.
N(A)={0}
e.
none of the above
[3 points]
10. Let A=
and b=
. Then the least square solution
to Ax=b is
8
a.
b.
c.
d.
e.
None of the above
[3 points]
11. Let A be an nxn matrix. Then, which one of the following statements is FALSE
a.
ATA is symmetric
b.
A+AT is symmetric
c.
A-AT is skew-symmetric
d.
ATA is invertible
e.
None of the above
[3 points]
12. If A is any 3x4 matrix, which one of the following statements is FALSE?
(a) The columns of A are linearly dependent.
(b) The row space of A is a subspace of
.
(c) The rows of the matrix form a basis for the row space of A.
(d) The system AX = 0 has infinitely many solutions.
(e) None of the above
[3 points]
13. Let T: V
W be a linear transformation and {v1,v2,,….vn} is a set of vectors in
V. Which one of the following statements is TRUE:
a.
If {v1,v2,,….vn} is linearly independent then {T(v1),T(v2,),….T(vn)} is linearly
independent in W
9
b. If {T(v1),T(v2,),….T(vn)} is linearly independent in W then {v1,v2,,….vn} is
independent in V
c.
If {v1,v2,,….vn} is a basis for V and T is onto then {T(v1),T(v2,),….T(vn)} is a
basis for W
d. If {v1,v2,,….vn} is a basis for V and T is one-to-one, then {T(v1),T(v2,),
….T(vn)} is a basis for W
e.
None of the above
[3 points]
14. If W is the subspace of
defined by
, then
=
a. 0
b. 1
c. 2
d. 3
e. None of the above
[3 points]
15. If T:
is a linear transformation such that T(v)
0 for some v in
, then
a. T is one-to-one.
b. T is onto.
c.
dim (N(T))=0.
d. T is invertible
e.
none of the above
[3 points]
PART III. Answer TRUE or FALSE only (Questions 16 to 27) IN THE
TABLE IN THE FRONT PAGE ( 2 points for each correct answer):
10
16. The set W of all skew-symmetric 3x3 matrices is a subspace of M3X3 isomorphic
to the vector space P2
17. If A is a 3x5 matrix with linearly independent row vectors, then nullity(A)=3
18. The matrices
and
are similar.
19. Any orthogonal set of n nonzero vectors in
forms a basis for
.
20. If W is a subspace of R4 (with the usual dot product) with dim W = 3, then W┴ is
isomorphic to R.
21. T: P2
R defined by T(ax2+bx+c)=
is a linear transformation
22. If A is an invertible 3x3 matrix, then A is diagonalizable
23. If λ=0 is an eigenvalue for an nxn matrix A, then A is not invertible.
24. if T:V
W is a linear transformation such that dim(V) = dim(W)=n and T is onto,
then T is 1-1
25. The set of all polynomials of degree 4 is a vector space under the usual addition and
scalar multiplication of polynomials.
26. Let A be a 44 matrix such that (rank A)= 4, then
AX=0 has a unique solution X=0.
27. Let A is a 3x3 nonzero matrix such that the non-homogeneous system Ax=b is
inconsistent then Rank (A)≠3
[24 points]
11