MATH 260 – Review for Final Exam
April 27, 2013
1. Let M pn,
Rq be the space of n-by-n matrices with real entries.
(a) Show that (with the operations of matrix addition and scalar multiplication), M pn,
vector space.
Rq is a
Rq? Exhibit a basis of M p3, Rq.
Suppose A is a fixed 3-by-3 matrix. Define the map A : M p3, Rq Ñ M p3, Rq by
(b) What is the dimension of M pn,
A pX q AX XA, so the operation A takes the difference between left multiplication by the fixed matrix
A and right multiplication by A.
(c) Show that A is a linear map from M p3,
Rq to M p3, Rq.
(d) Suppose A is an invertible matrix. Must A be invertible? If so, prove it. If not, what (with
proof) is the maximum possible value of the rank of A ? [Note: This problem is hard!]
2. Give examples of each of the following or explain why none can exist:
(a) A 2-by-2 matrix A such that A 0 but A2
0.
(b) A 2-by-2 matrix A such that A2 0 but A3 0.
(c) A 2-by-2 matrix A such that A1 4A.
¾
(d) A vector field F, defined and continuously differentiable for all px, y q
F nds π, where C is the unit circle x2
y2
P R2 ,
such that
1, traversed counterclockwise, n is the outward-
C
pointing normal vector to C, and ds is the element of arc length on C (this is the flux of F out of
C).
»»» (e)
B
A vector field F, defined and continuously differentiable for all px, y, z q
∇ FdV
3. For any z
4π, where B is the unit ball x2
a
ib P
(a) For z and w in
y2
z2
¤ 1.
C, and define the 2-by-2 matrix M pzq C, show that M pz
wq M pz q
a
b
b
P R3 ,
such that
a
M pwq and M pzwq M pz qM pwq.
(b) Show that M p0q is the zero matrix, M p1q is the identity matrix and M p1{z q pM pz qq1 .
Parts (a) and (b) show that the field of complex numbers is algebraically isomorphic to this set
of 2-by-2 matrices.
i2
(c) Recall that the quaternions H are defined as numbers of the form q a bi cj dk where
j 2 k2 1 and ij ji k, ki ik j and jk kj i. Find a mapping from H to a
2
H R
set of 4-by-4 matrices that is an isomorphism. (Hint: You know that 4 as real vector spaces,
so think about the linear mappings from
to
defined by [left] multiplication by i, j and k.)
H H
€ R3 be the graph of z f px, yq for px, yq in a region R in the xy-plane.
(a) Using the parametrization x u, y v, z f pu, v q, derive the formula
4. Let the surface S
areapS q »» a
1
R
}∇f }2 dx dy.
(b) Apply this formula to calculate the surface area of the part of the plane x
the first octant (x ¥ 0, y ¥ 0, z ¥ 0).
5. Let L be a linear transformation from
Lpr1, 1, 1, 1sq r1, 1s
2y
3z
6 in
R4 to R2 such that
Lpr1, 1, 0, 0sq r1, 0s ,
Lpr0, 1, 0, 1sq r0, 1s ,
Lpr0, 0, 0, 1sq r0, 0s.
(a) Calculate Lpr0, 0, 1, 1sq, Lpr0, 1, 0, 0sq, and Lpr1, 0, 0, 0sq.
(b) What is the rank of L? How do you know?
(c) What is the dimension of the kernel of L? How do you know?
(d) Find the matrix ML of L (with respect to the standard bases tr1, 0, 0, 0s, r0, 1, 0, 0s, r0, 0, 1, 0s, r0, 0, 0, 1su
of
R4 and tr1, 0s, r0, 1su of R2 .
(e) Let A
RART D.
ML pML qT . Find an orthogonal matrix R and a diagonal matrix D such that
6. Let f px, y, z q xy yz zx. Calculate the range of f (i.e., the minimum and maximum values)
on the set T tpx, y, z q | x ¥ 0 , y ¥ 0 , z ¥ 0 , x y z ¤ 1u.
7. If three resistors having resistances x, y and z are connected in parallel, the resistance R of the
entire circuit satisfies the equation
1
1
1 1
.
R
x y z
Use differentials to estimate R to three decimal places if x 2.012, y
»»
8. Evaluate the surface integral
S
, where S is the surface of the hemisphere x2
F p3xy
5z 2 q i
4.008 and z 3.992.
F n dσ
y2
p2y
4, z ¡ 0, where F is the vector field
ex q j p3x2 4y 2q k
z2
and n is the normal vector field to the surface that points away from the origin.
3
9. The two-dimensional wave equation describes the motion of a vibrating membrane (such as the
head of a drum). The function upx, y, tq gives the vertical (z-direction) displacement of the point
whose equilibrium position is px, y q at time t. The wave equation is
B2 u c2 ∆u
B t2
where c is a positive constant. Because the wave equation is second-order in t, a reasonable initial/boundary value problem for the wave equation:
B2 u c2 ∆u
Bt2
on D for all t ¥ 0
gives initial values for the position and velocity of the membrane:
upx, y, 0q f px, y q ,
Bu px, y, 0q gpx, yq
Bt
for px, y q P D
and boundary values
upx, y, tq hpx, y q
for px, y q P B D and all t ¡ 0.
By considering something related to the kinetic energy of the membrane:
»» Bu 2
E ptq Bt dx dy
1
2
D
show that there is at most one solution to this initial/boundary value problem.