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LINEAR ALGEBRA AND VECTOR ANALYSIS
MATH 22B
Unit 38: Final Exam Practice
Problems
Problem 38P.1) (10 points): a) Write down the Laplacian L of the complete graph with four vertices and find its eigenvalues and eigenvectors. b) Solve the wave equation ftt = −Lf on the graph with initial position f (0) = [2, 4 , 3 , 3].
Solution: a) The Laplacian is
L =
The eigenvalues are 0, 4 , 4 , 4 as one can see that the matrix A = L − 4 has 3 eigenvalues 0 and an eigenvalue −4. The eigenvectors are v 1 = [1, 1 , 1 , 1], v 2 = [− 1 , 1 , 0 , 0], v 3 = [− 1 , 0 , 1 , 0], v 4 = [− 1 , 0 , 0 , 1]. b) The initial condition is 3v 1 + v 2. We can write down the closed form solution f (t) = 3 cos(0t)[1, 1 , 1 , 1] + cos(2t)[− 1 , 1 , 0 , 0]
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Linear Algebra and Vector Analysis
Problem 38P.2) (10 points): a) The set of all 800 × 600 gray scale pictures are a subspace of a large vector space M (m, n) What is the dimension of this space? b) The map (x, y, z, w) → (y, z, w, x) defines a linear transformation on R^4. What is its determinant? c) The set of all words in the alphabet A − Z feature an addition + given by concatenation. The zero element 0 is the empty word. What is the name of this algebraic structure? d) Which of the three following people is not associated to an axiom system Euclid, P eano, Jordan. e) What does the rank-nullity theorem tell for a n × m matrix? f) If S is the eigenbasis of a n × n matrix A, what can you say about S−^1 AS? g) If A is a n × 1 matrix and A = QR is the QR-decomposition of A, what shape does the matrix R have? h) What is the name of a matrix A with complex entries for which A T A = 1. i) What is the determinant of a matrix A ∈ SU (2)? j) Which of the fundamental forces are associated to SU (2). The electromagnetic, the weak, the strong or the gravitational force?
Solution: a) 800 · 600 = 480′000. b) It is a permutation matrix with three upcrossings. The determinant is −1. c) It is a monoid. d) Jordan. e) The rank plus the nullity is is equal to m. f) It is a diagonal matrix. g) It is a 1 × 1 matrix. h) It is called unitary. i) 1, by definition. j) The weak force.
Linear Algebra and Vector Analysis
Problem 38P.4) (10 points): People on social media have been in war about expressions like 2 x/ 3 y − 1 if x = 9 and y = 2. Computers and humans disagree: most humans get 2, while most machines return 11. A psychologist investigates whether the size of the numbers influences the answer and asks people. This needs data fitting: using the least square method, find those a and b such that (^) ax
3 y
− b = 2
best fits the data points in the following table: x y 9 3 6 1 -3 1 0 1
Solution: We write down the systems of equations (don’t leave out any of the equations) a − b = 2 2 a − b = 2 −a − b = 2 −b = 2. The matrix is
A =
We have b =
. The solution^ ~x^ = (A
T (^) A)− (^1) AT (^) b is
[
]
. This is not a surprise as
a = 0, b = −2 exactly solves the problem.
Problem 38P.5) (10 points):
Let ~x =
v e f
(^) denote the number of vertices, edges and faces of a poly-
hedron. During a Barycentric refinement, this vector transforms as
A~x =
(^) ~x.
a) (5 points) Verify that ~v 1 =
, ~v 2 =
, and ~v 3 =
(^) are
eigenvectors of A and find their eigenvalues. b) (5 points) Write down a closed form solution of the discrete dynamical
system ~x(t + 1) = A~x(t) with the initial condition
v e f
Solution: a) Just compute A~v 1 and see that it is a multiple of ~v 1. Similarly, do that with A~v 2 and A~v 3. The eigenvalues are 1, 2 , 6. b) Write
(^) = c 1
(^) + c 2
(^) + c 3
giving c 1 = 1, c 2 = 3 and c 3 = 3. We can now write down the closed form solution
1 t
(^) + 2t^ · 3
(^) + 6t^ · 3
Problem 38P.6) (10 points): The Arnold cat map is T~v = A~v where
A =
[
]
It is an icon of chaos theory. a) (2 points) What is the characteristic polynomial of A? b) (2 points) Find the eigenvalues of A. c) (2 points) Find the eigenvectors of A. d) (2 points) Is the discrete dynamical system defined by A asymptotically stable or not? e) (2 points) Write down an orthogonal matrix S and a diagonal matrix B such that B = S−^1 AS.
Problem 38P.8) (10 points): Remember to give computation details. Answers alone can not be given credit. a) (2 points) The following matrix displays the solution of the Cellular automaton 10. Find its determinant
A =
b) (2 points) Find the determinant of
B =
c) (2 points) Find the determinant of
C =
d) (2 points) Find the determinant of
D =
e) (2 points) Find the determinant of E = 2Q + 5Q−^1 + 7I: (you can leave it in terms of eigenvalues of the basic circulant matrix Q you have seen. No simplifications are required):
E =
Linear Algebra and Vector Analysis
Solution: a) Patterns, 6 upcrossings, det= b) Patterns or row reduction det = −120. c) Partitioned. det = −450. d) Build B = A − 10 which has eigenvalues 0, 0 , 0 , 0 , 9, so that B has eigenvalues 10 , 10 , 10 , 10 , 19 which is 190000 e) The determinant is the product of the eigenvalues λ 1 λ 2 λ 3 λ 4 λ 5 where λk = 7 + 2 exp(2πik/5) + 5 exp(− 2 πik/5).
Problem 38P.9) (10 points): Find the general solution to the following differential equations: a) (1 point) f ′(t) = 1/(t + 1) b) (1 point) f ′′(t) = et^ + t c) (2 points) f ′′(t) + f (t) = t + 2 d) (2 points) f ′′(t) − 2 f ′(t) + f (t) = et e) (2 points) f ′′(t) − f (t) = et^ + sin(t) f) (2 points) f ′′(t) − f (t) = e−^3 t
Solution: a) log(t + 1) + C b) et^ + t^3 /6 + C 1 t + C 2. c) C 1 cos(t) + C 2 sin(t) + t + 2 d) C 1 et^ + C 2 tet^ + t^2 et/ 2 e) C 1 et^ + C 2 e−t^ + tet/ 2 − 1 /2 sin(t) f) C 1 et^ + C 2 e−t^ + e−^3 t/ 8
Linear Algebra and Vector Analysis
Solution: a) The equilibrium points are (0, 0) and (2, 2). b) The Jacobian matrices are [ 1 1 1 − 3
]
[
]
c) The first equilibrium point is unstable, the second is stable. d) We have phase portrait A.
Problem 38P.11) (10 points): a) (6 points) Find the Fourier series of the function which is 1 if |x| > 1 and −1 else. We call it the Pacific rim function.
f (x) =
1 |x| > 1 − 1 |x| ≤ 1
b) (4 points) Find the value of the sum of the squares of all the Fourier coefficients of f.
Solution: a) The function is even. It has a cos-series. We have
an =
π
0
− cos(nx) dx +
π
∫ (^) π
1
cos(nx) dx = −4 sin(n)/(πn).
and
a 0 =
π
0
2 dx +
π
∫ (^) π
1
2 dx
which is −
2 /π + (π − 1)
2 /π. It could be simplified to (π − 2)
2 /π. b) By Parseval, we know the sum of the squares
n=0 a
2 n equal to (2/π)^
∫ (^) π 0 1
(^2) dx = 2.
Problem 38P.12) (10 points): a) Solve the system ft = 3fxx − f + t with f (0, x) = x on [−π, π] b) Solve the system ft = 3fxx − 9 fxxxx with f (0, x) = x.
Solution: a) A particular solution which does not depend on x is f (t) = t^2 /2. Now look at the homogeneous part. Since the operator 3D^2 − 1 has eigenvalues λn = − 3 n^2 − 1, we have to look at the solution of ft = λf which has solution e−^3 n (^2) t
. The initial position is ∑^ ∞
n=
2(−1)n+ n
sin(nx).
The solution of the heat equation is now
fhom(t, x) =
∑^ ∞
n=
2(−1)n+ n
e(−^3 n
(^2) −1)t sin(nx).
The final solution is fpart(x, t) + fhom(t, x). b) Since the operator 3D^2 − 9 D^4 has eigenvalues λn = − 3 n^2 − 9 n^4 the solution is
∑^ ∞
n=
2(−1)n+ n
e(−^3 n
(^2) − 9 n (^4) )t sin(nx).
Problem 38P.13) (10 points): a) Solve the system ftt = 9fxx with f (0, x) = x and ft(0, x) = 11 sin(88x). b) Solve ftt = 9fxx − fxxxx with ft(0, x) = x.
