Australian Student Number: National
University
Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2016 MATH1116 — Advanced Mathematics and Applications 2 Book B — Linear Algebra
Exam Duration: 180 minutes. Reading Time: 15 minutes.
Materials Permitted In The Exam Venue:
• One A4 page with hand written notes on both sides. (This A4 page is to cover both Analysis and Algebra.)
• 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:
• 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 Algebra and Analysis sections are worth a total of 50 points each, with the value of each question as shown. It is recommended that you spend equal time on the Analysis and the Algebra papers.
• 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 10 pts
Let V be an inner product space. Suppose that v1, . . . , vn ∈ V are eigenvectors of a self- adjoint linear transformation T : V −→ V with distinct eigenvalues λ1, . . . , λn . Show that thevectors {v1,…,vn} arelinearlyindependentandpairwiseorthogonal.
Write your solution here
MATH1116 — Book B, Page 2 of 8
Question 2 7 pts
Suppose that T is a self-adjoint linear operator on a finite-dimensional inner product space V , and that 2 and 3 are the only eigenvalues of T . Prove that
T 2 − 5T + 6I = 0 .
Write your solution here
MATH1116 — Book B, Page 3 of 8
Question 3 7 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 — Book B, Page 4 of 8
Question 4 7 pts
Recall that the unit circle in C consists of the complex numbers λ such that λλ = 1. LetT :V −→V beinvertible,andassumethatT−1 =T∗.
Show that the eigenvalues of T all lie on the unit circle.
Write your solution here
MATH1116 — Book B, Page 5 of 8
Question 5
(a) Give the definition of an inner product on a vector space V . Write your solution here
(b) Prove that there exists a unique polynomial f ∈ P2 such that, for all д ∈ P2 , 1 1
д(t)cos(πt)dt = д(t)f(t)dt. −1
[Note: P2 denotes the vector space of polynomials of degree less than or equal to 2.]
Write your solution here
MATH1116 — Book B, Page 6 of 8
(c) What is the polynomial f in the previous part? 7 pts [The expression in your final answer is allowed to involve definite integrals; you do not have to compute such integrals.]
Write your solution here
MATH1116 — Book B, Page 7 of 8
Question 6 5 pts
Find the singular values of the matrix of the map T : R2 −→ R2 , where T(x,y) = (4x,3x−5y).
Write your solution here
MATH1116 — Book B, Page 8 of 8