Final Examination, Second Semester (November 2015)
Mathematics and Applications 2 Honours
one A4 sheet of handwritten notes (two-sided). u
(MATH1116)
Any application of the ANU logo on 3 hoaucrolsourdedubracktgirounn.d is subject
Writing period: Study period: Permitted materials:
Student Number:
to approval by the Marketing Office.
15 minutes duration. Please send to
Important Notes:
• No calculators or other electronic devices are permitted. No books are permitted.
• Check that you have a separate page with a diagram, which is needed
for Q9. Anything written on that separate page will not be marked.
• Write your student number in the space provided above.
• Write your answers in the spaces provided. If you need more space, use the back of the page the question is on. There is also a blank extra page provided at the back of this booklet. Make a note on the first page of any questions for which you have used extra pages.
• You must prove or justify your answers, except for questions which explicitly state otherwise. Do not expect credit for a correct answer with no justification. Write clearly and legibly.
• The questions are each worth varying marks, split into parts; the value of each part is shown.
• A good strategy is not to spend too much time on any question. Read the questions through first and attack them in the order that allows you to make the most progress.
• There are 100 marks available on this exam: the linear algebra questions total 50 marks and the analysis questions total 50 marks.
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Total
MATH1116 Final Exam, November 2015 Page 1 of 16
Question 1 6 marks For each of the following, state whether or not the given proposition is ‘True’ or ‘False’.
You do not need to justify your answers for this question. (Each part has equal weight.)
a) There exists a one-to-one linear function from R2 to R.
b) There exists an isomorphism from P2 to R3. (Recall here that P2 is the vector space of real polynomials of degree less than or equal to 2.)
c) There exists an R-linear map from C to itself which is not C-linear.
d) There exists a diagonalizable 2 by 2 real matrix with 3 non-zero entries whose
eigenvalues are 4 and 5.
e) There exists a 2 by 2 real matrix with eigenvalues 0 and 1 which cannot be diago- nalized.
f) There exist 3 linearly independent vectors which span R2. Question 2
a) For each of the following three questions, answer ‘Yes’ or ‘No’. You do not need to justify your answers for this question.
i) Suppose that f : R2 ! R has the property that limf(x,0) = limf(0,y) = f(0,0).
x!0 y!0 Must f be continuous at (0,0)?
ii) Suppose that g : R2 ! R has the property that there exists M 2 R such that,
for all (x,y) 2 R2,
Must g be continuous at (0,0)?
|g(x,y)| Mk(x,y)k.
iii) Supposethath:R2 !Rissuchthatforally:R!Roftheformy(x)=ax
(where a 2 R),
lim h(x, y(x)) = h(0, 0) . Must h be continuous at (0,0)?
MATH1116 Final Exam, November 2015 Page 2 of 16
(Question 2 Cont.) b) Suppose that:
• U = R with distance given by the absolute value function;
• V = Rn with the Euclidean norm;
• W is a finite dimensional normed vector space;
• X is an infinite dimensional normed vector space; and
• Y is an arbitrary metric space.
i) List those spaces out of U, V, W, X and Y in which it must be the case that
Cauchy sequences converge.
You do not need to justify your answer for this question. (2 marks)
ii) List those spaces out of U, V, W, X and Y in which it must be the case that convergent sequences are Cauchy.
You do not need to justify your answer for this question.
c) Suppose that f : Rn ! R is continuous on a closed rectangle R = Circle all of the following that must be true.
You do not need to justify your answers for this question.
• f is bounded on R.
• f attains maximum and minimum values on R. • f is uniformly continuous on R.
• f is di↵erentiable on R.
• f is integrable on R.
[aj,bj] ⇢ Rn. (3 marks)
MATH1116 Final Exam, November 2015
Page 3 of 16
Question 3 10 marks
I have two kinds of food in my o ce – 85% extra dark chocolate, and peanut M&M’s. Each hour I take a bite of one of the two, and which one I eat depends only on what I ate during the previous hour. The transition matrix for my choice is
P=.8 .1 . .2 .9
(In the rows and columns of P , the first basis vector corresponds to 85% extra dark chocolate, and the second basis vector corresponds to peanut M&M’s.)
Assuming I live forever — which I surely will on this diet — what percentage of my bites will be M&M’s?
MATH1116 Final Exam, November 2015 Page 4 of 16
Question 4
a) Determine whether the series P1 n2 converges or diverges.
Justify your answer using appropriate theorems / tests.
b) Determine whether the series P1 ne n2 converges or diverges. n=1
Justify your answer using appropriate theorems / tests.
MATH1116 Final Exam, November 2015
Page 5 of 16
(Question 4 Cont.)
c) Determine whether the series P1 cos(n) converges or diverges. Justify your answer
using appropriate theorems / tests. (3 marks)
MATH1116 Final Exam, November 2015 Page 6 of 16
Question 5 10 marks
a) Give an example of an inner product on R2 which is not the dot product on R2. (Make sure you prove that your example satisfies the axioms of an inner product.) (5 marks)
MATH1116 Final Exam, November 2015 Page 7 of 16
(Question 5 Cont.)
b) Let P2 denote the real vector space of all polynomials of degree at most 2, with
inner product given by
hp, qi = p(t)q(t)dt.
Starting with the basis {1,1+t,t2}, use Gram-Schmidt to construct an orthonormal
basis of P2. (5 marks)
MATH1116 Final Exam, November 2015 Page 8 of 16
Question 6
We consider the power series defined by
converges.
f (x) = ln 2 +
a) Determine the set of real numbers x for which ln 2 + P1 xn
For which x does the series converge absolutely? For which x does the series converge conditionally?
MATH1116 Final Exam, November 2015
Page 9 of 16
(Question 6 Cont.)
Recall that we are considering the power series defined by
f (x) = ln 2 + 2n n .
b) Prove that f0(x) = 1 , for suitable values of x.
c) Prove that ln 2 = P1 ( 1)n . 3 2nn
MATH1116 Final Exam, November 2015
Page 10 of 16
Question 7 12 marks
Let v1, . . . , vk be non-zero eigenvectors of a real symmetric n by n matrix A with distinct eigenvalues. Show that the eigenvectors v1, . . . , vk are linearly independent and orthogonal to each other.
MATH1116 Final Exam, November 2015 Page 11 of 16
Question 8
a) Letg:[1,e]⇥[0,1]!Rbedefinedby
g(x, y) = yxy + xy .
Explain why g is integrable, and show that
[1,e]⇥Z[0,1]
g(x,y)d(x,y) = e2 e 1.
MATH1116 Final Exam, November 2015
Page 12 of 16
b) Compute
Z 1 ✓Z 1 ◆ sin(y2)dy
0x justifying the steps taken in your calculation.
MATH1116 Final Exam, November 2015
Page 13 of 16
Question 9 — See diagram provided separate to exam booklet 12 marks Let A be a real 5 by 7 matrix with rank 3. Denote by {v1,…,v7} the columns of V and
by {u1, . . . , u5} the columns of U in a singular value decomposition A = U⌃V T
ofA,suchthatvi istheithcolumnofV andui istheithcolumnofU.
The 4 squares in the attached diagram correspond to subspaces; the rectangles labeled L1 and L2 on the left are subspaces of R7, and the subspaces R1,R2 on the right are subspaces of R5. Each of the basis vectors ui, vi has been placed in the rectangle labelling the subspace to which it belongs (for example, the vector v3 belongs to the subspace L1).
For each of the vectors described below, say to which of the subspaces this vector necessarily belongs. If the vector described is necessarily the 0 vector, say that it is the 0 vector.
That is, write ‘L1’, ‘L2’, ‘R1’, ‘R2’, or ‘0’, in each of the spaces provided below.
You do not need to justify your answers for this question. (Each part has equal weight.)
a) The first column of A:
b) A vector x in the nullspace of AT :
c) A vector y which is orthogonal to every vector in the row space of A: d) A vector z in the nullspace of AT A:
e) The vector Av2: f) The vector AT u4:
g) A vector w which is orthogonal to every vector in the nullspace of AT : h) A linear combination of u4 and u5 in the column space of A:
MATH1116 Final Exam, November 2015
Page 14 of 16
Question 10 10 marks
The following result generalises one of the convergence tests studied this semester. It is not su cient to rely on that particular test in answering this question.
You must provide a full proof.
denotes the supremum, or least upper bound, of the
N!1 n N an equals `, for some ` 2 [0, 1). Prove that P1 an converges.
Recall that, for each N 2 N, sup an+1
n N an set{an+1 :n N,n2N}.
Let (an)1n=1 be a positive sequence of real numbers, such that lim ✓sup an+1 ◆ exists and
MATH1116 Final Exam, November 2015 Page 15 of 16
This page has been intentionally left blank, to be used for extra writing space if required.
MATH1116 Final Exam, November 2015 Page 16 of 16