# CS代考 MATH1116 — Advanced Mathematics and Applications 2 Book B — Algebra – cscodehelp代写

Australian Student Number: National

University

Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2017 MATH1116 — Advanced Mathematics and Applications 2 Book B — 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 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

LetT :C4 −→C4 begivenby

T(z1,z2,z3,z4) = (0,3z1,2z2,−3z4).

Find the eigenvalues of T and the singular values of T (with multiplicity). Write your solution here

MATH1116 — Book B, Page 2 of 10

Question 2 10 pts

(a) Suppose U is the subspace of R4 defined by

U =span{(6,2,−2,−2),(−1,1,−1,−1)}.

Find an orthonormal basis of U and an orthonormal basis of U ⊥ . 5 pts Write your solution here

MATH1116 — Book B, Page 3 of 10

(b) WhathappenswhentheGram-Schmidtalgorithmisappliedtoalinearlydependent list of vectors? Explain. 5 pts

Write your solution here

MATH1116 — Book B, Page 4 of 10

Question 3 9 pts

For each matrix below, write Yes if the matrix is diagonalizable over C, and write No otherwise. Provide a short justification for your answer. (The justification does not have to be a computation!)

(a) 0 2 2

(b) 0 1 0

(c) 0 1 1

(d) −1 0 0

(e) −3 2 0

(f)3−i 2 0

MATH1116 — Book B, Page 5 of 10

Question 4 9 pts

Let T : V −→ W be a linear map between finite-dimensional vector spaces.

(a) Give the definition of the dual map T ′ of T . (Make sure to state clearly what the

domain and codomain of T ′ are.) 2 pts Write your solution here

(b) Prove that T is injective if and only if T ′ is surjective. 7 pts Write your solution here

MATH1116 — Book B, Page 6 of 10

Extra space for previous question

MATH1116 — Book B, Page 7 of 10

Question 5 14 pts

Let V be a finite-dimensional complex inner product space.

(a) GiveanexampleofanoperatorT :V −→V suchthatT9 =T8,butT2 T. 3pts

Write your solution here

MATH1116 — Book B, Page 8 of 10

(b) SupposenowthatT isnormal,andthatT9 =T8.ShowthatT2 =T. 7pts [Hint: Spectral theorem.]

Write your solution here

MATH1116 — Book B, Page 9 of 10

(c) Is the T in part (b) above necessarily self-adjoint? Explain. 4 pts Write your solution here

MATH1116 — Book B, Page 10 of 10