MATH 601: Methods of Applied Mathematics I
Instructor:
Prof. Wolfgang Bangerth
bangerth@math.tamu.edu
Midterm exam 10/22/2012
Below are eight problems that you should work on. If you can see that you -are running out of time, do the
simpler questions rst.
MATH 601: Quiz 10 (11/14/2012)
Name:
UIN:
Problem 1 (5 points): Consider the matrix A =
2
1
1
2
and answer the following questions:
What is the characteristic polynomial of this matrix?
Answer: The characteristic polynomial is dened as the determinant of
MATH 601: Quiz 9 (11/07/2012)
Name:
UIN:
Problem 1 (5 points): For each of the following mappings, state whether it is linear or not. If it is not
linear, explain why.
v1
v1 + 1
F : R3 R2 dened by
F (v ) :=
where v = v2 .
v2 + 1
v3
Answer: This mapping i
MATH 601: Quiz 8 (10/31/2012)
Name:
UIN:
Problem 1 (5 points): Consider the following set of vectors in V = R3 :
1
0
0
1
u1 = 0 , u2 = 2 , u3 = 3 , u4 = 0 .
0
0
4
5
Answer these questions:
Are these vectors linearly independent? If not, why not?
Answer:
MATH 601: Quiz 7 (10/17/2012)
Name:
UIN:
We call a set V a vector space over the scalar eld K if
a vector addition is dened so that u, v V : u + v V
a scalar multiplication is dened so that u V, k K : ku V
and in addition these two operations satisfy th
MATH 601: Quiz 6 (10/08/2012)
Name:
UIN:
2
Problem 1 (5 points): Consider the matrix A = 1
0
1
2
1
0
1. Compute its inverse.
2
Answer: Starting with the linear system with multiple right hand sides,
AX = I
we can do the forward elimination and backward su
MATH 601: Quiz 5 (10/03/2012)
Name:
UIN:
1a
1
x=
. For which values of the parameter
a9
2
a does the linear system have a unique solution? For which values of a (if any) does it have innitely many
solutions? For which values of a (if any) does it have no
MATH 601: Quiz 3 (9/19/2012)
Name:
UIN:
Problem 1: For the following two matrices, do the following: (i) State whether it is invertible. If it is in
fact invertible, then also: (ii) Show the inverse A1 of the matrix; (iii) verify that it is indeed the inv
MATH 601: Quiz 3 (9/19/2012)
Name:
UIN:
Problem 1: Let
1
A = 3
0
2
0 ,
1
B=
1
0
2
1
3
,
2
1
u = 1 ,
1
1
v = 2 .
3
For the following statements, determine whether they are mathematically valid and if so, compute their
value:
147
1. AB = 3 6 9
012
2. B T A:
MATH 601: Quiz 2 (9/10/2012)
Name:
UIN:
Problem 1: Find a parametric representation (using a parameter t) of a line that goes through points
P = (1, 1) and Q = (3, 1).
Answer: The located vector pointing from P to Q is given by P Q= Q P = (2, 0). Then, ac
MATH 601: Quiz 1 (9/5/2012)
Name:
UIN:
Problem 1: Show that for any two vectors u, v Rn there holds that
proj(u, v ) = proj(u, v ),
i.e., that the projection of u onto the direction of v is the same as the projection of u onto the direction of
the directi
MATH 601: Quiz 11 (11/28/2012)
Name:
UIN:
Problem 1 (4 points): Consider the function f (z ) = z where the complex conjugate z of a number
z = x + iy C is dened as z = x iy . Answer the following questions:
Is f (z ) linear? If not, why not?
Answer: No.