12
The Wave Equation
d2 u d2 u - 2 = 0, dt2 dx
The wave equation in one space and one time variable is u : (Rx , Rt ) R
Theorem. If f0 , f1 C (Rx ), !u C (R2 ) satisfying the wave equation with the initial x,t conditions u(0, x) = f0 (x) and x u(0, x) = f
xercises on solving
x = 0: pivot variables, special solutions
Problem 7.1:
a) Find the row reduced form of:
2
1
4 0
=
2
5 7
4 1
2 11
b) What is the rank of this matrix?
c) Find any special solutions to the equation
3
9
7 5
3
x = 0.
Solution:
a) To transfo
Part Problems and Solutions
Problem 1: Find the general solution by separation of variables:
dy
=2
dx
Solution:
dy
2
ln |2
IC: y(0) = 2
y
= dx !
y | = x + c ! |2
y=2
=0!
e
y (0) = 0
y,
x
Z
dy
2
y
y| =
(with
= 2 ! y = 2(1
e
=
x
e
Z
dx !
(with
= ec ) !
any
xercises on the geometry of linear equations
Problem 1.1: (1.3 #4. Introduction to Linear lgebra: Strang) Find a combination x1 w1 + x2 w2 + x3 w3 that gives the zero vector:
2 3
2 3
2 3
1
4
7
4 2 5 w2 = 4 5 5 w3 = 4 8 5 .
w1 =
3
6
9
Those vectors are (in
Exercises on factorization into
= LU
Problem 4.1: What matrix E puts into triangular form E
tiply by E 1 = L to factor into LU.
= U? Mul-
2
3
1 3 0
=4 2 4 0 5
2 0 1
Solution: We will perform a series of row operations to transform the
matrix into an upper
xercises on matrix spaces; rank 1; small world graphs
Problem 11.1: [Optional] (3.5 #41. Introduction to Linear lgebra: Strang)
Write the 3 by 3 identity matrix as a combination of the other ve permutation matrices. Then show that those ve matrices are li
Part
Problems and Solutions
Problem 1: [Natural growth, separable equations] In recitation a population model was
studied in which the natural growth rate of the population of oryx was a constant k > 0,
so that for small time intervals Dt the population c
18.06SC Final Exam Solutions
1 (4+7=11 pts.)
Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions:
1
2
1
1
x1 =
and
x2 =
1
0
0
1
(a) Remembering that A is 3 by 4, nd its row reduced echelon form R.
(b) Find the dimensions of all
Differential Equations
1. Denition of Differential Equations
differential equation is an equation expressing a relation between a
function and its derivatives. For example, we might know that x is a function of t and
.
.
x + 8x + 7x = 0.
(1)
or perhaps th
18.06SC Unit 3 Exam Solutions
1 (34 pts.)
(a) If a square matrix A has all n of its singular values equal to 1 in the
SVD, what basic classes of matrices does A belong to ? (Singular,
symmetric, orthogonal, positive denite or semidenite, diagonal)
(b) Sup
Variables and Parameters
1. Independent and
ependent Variables
When we write a function such as
f ( x ) = 3x2 + 2x + 1
we say that x is an independent variable: it can be freely set to any value
(or any value within the given domain) and the value of the
Exercises on column space and nullspace
Problem 6.1: (3.1 #30. Introduction to Linear
and T are two subspaces of a vector space V.
a)
lgebra: Strang) Suppose S
enition: The sum S + T contains all sums s + t of a vector s in S and
a vector t in T. Show tha
18.06SC Unit 2 Exam Solutions
1 (24 pts.)
Suppose q1 , q2 , q3 are orthonormal vectors in R3 . Find all possible values
for these 3 by 3 determinants and explain your thinking in 1 sentence each.
(a) det q1 q2 q3
(b) det q1 + q2
=
q2 + q3
(c) det q1 q2 q3
xercises on transposes, permutations, spaces
Problem 5.1:
(2.7 #13. Introduction to Linear lgebra: Strang)
a)
ind a 3 by 3 permutation matrix with P3 = I (but not P = I).
b)
b
b
ind a 4 by 4 permutation P with P4 6= I.
Solution:
a) Let P move the rows in
xercises on elimination with matrices
Problem 2.1: In the two-by-two system of linear equations below, what
multiple of the rst equation should be subtracted from the second equation when using the method of elimination? onvert this system of equations to
xercises on graphs, networks, and incidence matrices
Problem 12.1: (8.2 #1. Introduction to Linear lgebra: Strang) Write down
the four by four incidence matrix for the square graph, shown below.
(Hint: the rst row has -1 in column 1 and +1 in column 2.) W
to finish our proof we must show that is a countably additive measure Lemma. is countably additive on MF Proof. Want to show that Bi MF , Bi Bj = , i = j (B) = (Bi )
i=1 We can easily prove inequality one way.
BN =
N
i=1
i=1
Bi = B MF , then
Bi = BN B