Math 601 Midterm 2 Sample
This exam has 10 questions, for a total of 100 points + 5 bonus points.
Please answer each question in the space provided. You need to write full solutions.
Answers without justication will not be graded. Cross out anything the g
Spring 2014
Math 601
Name:
Quiz 4 sample
Question 1. (10 pts)
Determine whether the following statements are true or false. If false, explain why.
(a) If a vector space V has dimension n, then any n + 1 vectors in V must span V .
Solution: False. For exam
Spring 2014
Math 601
Name:
Quiz 3 sample
Question 1. (10 pts)
Determine if the given subset is a subspace of the corresponding vector space. (Show
work!).
(a) (5 pts) The subset of R3 :
W = cfw_(x, y, z) R3 | x + y + z 0
Solution: Consider u = (1, 1, 1).
Spring 2014
Math 601
Name:
Quiz 1 sample
Question 1. (12 pts)
(a) (5 pts) Find equations of the line L that passes through the points A(1, 0, 4, 3) and
B(3, 2, 0, 1).
Solution: First, calculate the direction of the line:
#
AB = (2, 2, 4, 2)
Then the equa
Math 601 Midterm 1 Sample
Name:
This exam has 9 questions, for a total of 100 points.
Please answer each question in the space provided. You need to write full solutions.
Answers without justication will not be graded. Cross out anything the grader should
Spring 2014
Math 601
Name:
Quiz 5 Sample
Question 1. (10 pts)
Let F : R2 R2 be the linear transformation dened by F (x, y) = (2x+3y, 4x5y). Find
the matrix representation of F with respect to the basis S = cfw_u1 , u2 = cfw_(1, 2), (2, 5).
Solution: This
Spring 2014
Math 601
Name:
Quiz 9 sample
Question 1. (10 pts)
Evaluate the integral
C
ez
dz
z 2 4z + 3
(a) when C is the circle |z 3| = 1, that is, the circle centered at 3 with radius 1.
Solution: Notice that
ez
ez
=
z 2 4z + 3
(z 1)(z 3)
So z = 3 is ins
Spring 2014
Math 601
Name:
Quiz 8 sample
Question 1. (10 pts)
(a) Determine whether the function f (z) = |z|2 is analytic on C.
Solution: f (z) = x2 +y 2 . So the real part is u(x, y) = x2 +y 2 and the imaginary
part is v(x, y) = 0. We have
u
= 2x,
x
u
=
Spring 2014
Math 601
Name:
Quiz 6 Sample
Question 1. (5 pts)
Verify that the rotation matrix A =
cos sin
is an orthogonal matrix.
sin cos
Solution: By denition, we only need to check that
AAT = AT A = I2
This is a straightforward calculation. I leave it