MATH1116 — Advanced Mathematics and Applications 2 ALL PARTS
Mathematical Sciences Institute EXAMINATION: End of Semester, Semester 2, 2020
Duration (combined for Algebra and Analysis, Parts A – F): 200 minutes. Materials Permitted And General Conditions:
• As per Exam Conditions document posted on Wattle and sent by email.
• Any resources not listed as permitted in the exam conditions are explicitly forbidden, and use of unauthorised resources will be considered academic misconduct.
Instructions To Students:
• Unless specically noted otherwise, you must prove your answers.
• Please be neat. Illegible answers may not receive full credit.
• You will be evaluated on the quality of your exposition as well as the correctness of your answer.
• If you use a result proved elsewhere, state the result clearly before applying it.
• After the 200 minutes has nished, please upload your answers to the relevant Wattle tool as soon as possible.
• You must stop writing after 200 minutes. If you have technical problems upload- ing your answer, take a picture and email it to Joan and Gri to establish a timestamp. You may then upload the same solution later in the afternoon. You may make no further changes to the solution, and you must follow up with an email explaining the issue so that the late submission will be accepted.
Question 1 (Upload to Part A Submission Tool) 9 pts
For each part below, give an example of the object(s) described or state that no such object exists. In either case, justify your answer.
(a) Operators R,T 2 L(V ) with the same characteristic polynomial which are not similar.
8> 26 12 37 26 12 37 26 12 37 9> (b) An orthonormal basis for span <> 6 0 7 , 6 3i 7 , 6 3i 7 => .
:64 0 75 64 0 75 64 4 75>; (c) An operator on R4 with no eigenvectors.
MATH1116 End of Semester Exam, 2020 PART A, Page 1 of 3
Question 2 (Upload to Part A Submission Tool) 6 pts
*. 2 a 0 +/
LetA=.0 2 0/bethematrixforT2L(V)withrespecttothestandardbasisonC3.
,00b-
For each question below, describe the conditions on the scalars a,b which ensure T has the
given property. You do not need to justify your answer. (a) T is diagonalisable.
(b) T is invertible.
MATH1116 End of Semester Exam, 2020 PART A, Page 2 of 3
Question 3 (Upload to Part A Submission Tool) 9 pts
Consider the vectors u = 26 1 37, = 26 1 37, w = 26 0 37, x = 26 4 37. Note that B = {u,} isabasisforR2. 64275 64 1 75 64375 64575
(a) Express w in B coordinates.
(b) Dene an inner product so that {u,} form an orthonormal basis for R2 . With respect to your newly-dened inner product, compute hw,xi.
(c) Is it possible to dene an inner product so that {u,,w} form an orthonormal set? Why or why not?
MATH1116 End of Semester Exam, 2020 PART A, Page 3 of 3
Question 4 (Upload to Part B Submission Tool) 7 pts
Let V be a 5-dimensional vector space. Show that if every 3-dimensional subspace of V is T -invariant, then T is a scalar multiple of the identity.
MATH1116 End of Semester Exam, 2020 PART B, Page 1 of 2
Question 5 (Upload to Part B Submission Tool) 6 pts
Suppose that V is a nite-dimensional inner product space.
Suppose that R 2 L(V) is a self-adjoint operator such that R6 = R5. Show that R2 = R.
MATH1116 End of Semester Exam, 2020 PART B, Page 2 of 2
Question 6 (Upload to Part C Submission Tool) 7 pts
Let V be an n-dimensional vector space and suppose that T : V ! V has n distinct eigenvalues. Show that if W ⇢ V is a T -invariant subspace, the restriction T |W : W ! W is necessarily diagonalisable.
MATH1116 End of Semester Exam, 2020 PART C, Page 1 of 2
Question 7 (Upload to Part C Submission Tool) 6 pts
26 0 0 a 0 37
SupposethatA=60 0 0 b7wherebandcarepositiverealnumbersandaisa
4c0005
negative real number. Assume that |a|, |b|, and |c| are distinct.
Compute a singular value decomposition of A.
MATH1116 End of Semester Exam, 2020 PART C, Page 2 of 2
Question 8 (Upload to Part D Submission Tool) 11 pts
Let:R2 !Rbegivenby
(x,) =x4 2×2 3 +3.
Find all the critical points of , and say which type (local minimum, local maximum, or saddle point) each critical point is.
[Make sure to show all of your working. As with all questions, answers that are not fully justied will not receive full credit.]
MATH1116 End of Semester Exam, 2020 PART D, Page 1 of 2
Question 9 (Upload to Part D Submission Tool)
5 pts
Let x : [0, 2020] ! R3 be dened by
x (t ) = ✓3 cos t , 3 sin t , 4t ◆ .
555
(a) Find the length of the curve C = {x(t) : t 2 [0,2020]}.
(b) Is x the arc length parametrisation of C ?
MATH1116 End of Semester Exam, 2020
PART D, Page 2 of 2
Question 10 (Upload to Part E Submission Tool)
20 pts
(a) (i) Find the set S of real numbers x for which the series X1 n4n1xn1 converges. n=1
(ii) For which x does the series converge absolutely? (iii) For which x does the series converge conditionally?
[Question continues on next page.]
8 pts
MATH1116 End of Semester Exam, 2020 PART E, Page 1 of 3
(b) With S being the set (the interval of convergence) you found in Part (a), dene
f:S!Rby X1 n1 n1 f(x)= n4 x .
n=1
(i) Determine a power series representation for
(ii) Calculatethevaluesofeachof f0(1/8), f(1/8),and [Part (b) continues on next page.]
Rx 0
R1/8 0
f (t ) dt .
2 pts f(t)dt. 6pts
MATH1116 End of Semester Exam, 2020 PART E, Page 2 of 3
(iii) (Part (b) continued). In this particular case, you should have that the interval of
Rx 0
• for each x 2 S nd an appropriate value x0 2 S to explicitly show that the convergence of the given series for f (x0) implies the convergence of the
Rx 0
convergence of the given series for f (x ) .
convergence of the power series you have found for
set S . Demonstrate the proof that the radius of convergence of a power series does not change when integrated or dierentiated, using this specic case as an example. What you are expected to do is:
series you have found for
• for each x 2 S nd an appropriate value x0 2 S toRexplicitly show that
f (t ) dt , and
the convergence of the series you have found for f (t ) dt implies the
f (t ) dt is the same as the
x0 0
4 pts
MATH1116 End of Semester Exam, 2020 PART E, Page 3 of 3
Question 11 (Upload to Part F Submission Tool) 14 pts
This question has two parts (Part (a) on this page, Part (b) on the next page), which are in no way related to one another. The 14 points available for this question will be divided as follows: your best result out of the two parts will be marked out of 10 points, and the other part will be marked out of 4 points.
(a) Suppose Rn is made into a real inner product space by giving it the standard dot product as its inner product h·, ·i , such that the standard Euclidean norm is the norm.
Let T be a self-adjoint linear operator on Rn , and suppose p 2 Rn . Dene a function
f : Rn ! R by
Show that f is dierentiable at p , with
f (x) = hTx,xi. dfp() = 2hTp,i
for all 2 Rn .
Note that you may assume the result, from the workshops, that T must be continuous.
MATH1116 End of Semester Exam, 2020 PART F, Page 1 of 2
(b) Suppose that f : Rn ! R is both continuous and such that lim f (x ) exists.
kxk!1
Denition: the existence of lim f (x ) is dened to mean that there is a value L 2 R
Prove that f is uniformly continuous. kxk!1
so that for all > 0 there exists M 2 R such that, for x 2 Rn , if kx k M then |f(x)L| <.
Note that you are allowed to use the theorem that if a function is continuous on a closed and bounded subset of Rn , then it is uniformly continuous on that subset.
MATH1116 End of Semester Exam, 2020 PART F, Page 2 of 2