Australian Student Number: National
University
Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2018 MATH1116 — Advanced Mathematics and Applications 2 Book B — Linear Algebra
Exam Duration: 180 minutes. Reading Time: 15 minutes.
Materials Permitted In The Exam Venue:
• Unmarked English-to-foreign-language dictionary (no approval from MSI required). • No electronic aids are permitted e.g. laptops, phones, calculators.
Materials To Be Supplied To Students:
• Formula Sheets Booklet (related to Analysis half of exam only). • Scribble Paper.
Instructions To Students:
• Answer the Analysis questions in Book A, and the Algebra questions in Book B, in the spaces provided. If you run out of space, you may use the backs of pages, but please make a note on the front of the page that your solution is continued on the back.
• The value of each question is as shown. It is recommended that you spend equal time on the Analysis and the Algebra papers (50 points each).
• A good strategy is not to spend too much time on any question. Read them through first and attack them in the order that allows you to make the most progress.
• You must prove / justify your answers, unless explicitly told otherwise. Please be neat.
Total / 50
Question 1 8 pts (a) Givethedefinitionofaself-adjointoperatoronarealinnerproductspace(V,⟨•,•⟩).
2 pts (b) State the spectral theorem for such operators. 2 pts
(c) Let T : R3 −→ R3 be self-adjoint, and suppose that T (1, 1, 1) = (2, 2, 2) . Suppose that (x,y,z) ∈ null(T). Show that x +y +z = 0. 4 pts
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 2 of 10
Extra space for previous question
MATH1116 End of Semester Exam, 2018 — Book B, Page 3 of 10
Question 2 8 pts
Let U be the subspace of R4 spanned by (1,1,0,0) and (1,1,1,2).
Find u ∈ U such that the distance from u to the point (1, 2, 3, 4) is as small as possible.
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 4 of 10
Question 3 8 pts
Let V be a finite-dimensional complex vector space, and let T : V −→ V be a linear map. Defineafunction f :C−→R by
f (λ) = dim range(T − λ id).
Prove that f is not a continuous function. Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 5 of 10
Question 4 8 pts
(a) Find the singular values of the standard matrix of the map T : R2 −→ R2 , where T(x,y) = (4x,3x−5y).
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 6 of 10
(b) LetT :V −→V bealinearmap. ShowthatT isinvertibleifandonlyif0isnota singular value of T . 4 pts
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 7 of 10
Question 5 8 pts
Suppose that V is a finite-dimensional inner product space, and let T : V −→ V be an invertible linear map satisfying T −1 = T ∗ . Show that the eigenvalues of T lie on the unit circle in the complex plane.
[Hint: recall that a complex number λ lies on the unit circle when λλ = 1.]
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 8 of 10
Question 6 10 pts
Suppose that on day 0, Andrew has a0 twizzlers and Barbara has b0 twizzlers. Each day, Andrew and Barbara exchange some twizzlers: Andrew gives 13 of his twizzlers to Barbara and keeps the other 23 ; Barbara, not to be outdone, gives 12 of her twizzlers to Andrew, and keeps the other half for herself. Neither Andrew nor Barbara actually enjoy twizzlers, so neither one ever eats any. Millions of years from now, when Andrew and Barbara are still alive and still trading twizzlers, what fraction of the a0 + b0 total twizzlers will Andrew have? How does this answer depend on the starting values a0, b0 ? Make sure you justify your claims rigorously.
[Hint: Denote by an the number of twizzlers Andrew has on day n, and bn the number of twizzlers Barbara has on day n. Consider a 2-by-2 matrix M such that for all n ≥ 1,
a 0 a n Mn = .
b0 an
What linear algebra tools do you have at your disposal to study what happens as n → ∞?] Write your solution here
MATH1116 End of Semester Exam, 2018 — Book B, Page 9 of 10
Extra space for previous question
MATH1116 End of Semester Exam, 2018 — Book B, Page 10 of 10