Student Number:
Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2019 MATH1116 — Advanced Mathematics and Applications 2 Book B — 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:
• Scribble paper. • Formula sheet.
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 marks for the questions in each book sum to 50.
• Unless specically noted otherwise, you must prove your answers.
• Please be neat. Illegible answers may not receive full credit.
Total / 50
Question 1 15 pts
For each part below, give an example of the object(s) described or state that no such map exists. In either case, justify your answer.
(a) Operators R,T 2 L(V ) with the same eigenvalues, but such that there is no invertible S withR=S 1TS.
Write your solution here
(b) An operator M 2 L (W ) which is diagonalisable but not invertible. Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 2 of 10
For each part below, give an example of the object(s) described or state that no such map exists. In either case, justify your answer.
(c) An operator M 2 L (Y ) which is invertible but not diagonalisable. Write your solution here
(d) An operator Q 2 L (U ) with at least 3 distinct square roots. Write your solution here
(e) A non-zero nilpotent operator N 2 L(Z). Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 3 of 10
Question 2
For this question, dene an inner product on R3 by h(a1,a2,a3),(b1,b2,b3)i=2a1b1 +a2b2 +a3b3
U =span{(1,1, 2),( 1,1,1)}. Find an orthonormal basis for U .
Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 4 of 10
Question 3 6 pts
Each matrix below describes an operator on C3 with respect to the standard basis.
Write Yes if the given matrix is diagonalisable over C, and write No otherwise. Provide a short justication for your answer.
*. 1 2 0 +/ (a).0 1 0/
Write your solution here
*. 1 0 0 +/ (b).0 1 1/
Write your solution here
*. 2 3 0 +/ (c). 3 2 0/
Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 5 of 10
Question 4 12 pts
Let V be a nite dimensional inner product space and let T 2 L (V ) . Suppose that U ⇢ V is a T -invariant subspace.
(a) Show that if T is self-adjoint, then U ? is also a T -invariant subspace. Write your solution here
(b) Show by example that if T is not self-adjoint, then U ? need not be T -invariant. Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 6 of 10
Recall that V is a nite dimensional inner product space and T 2 L(V). Suppose that U ⇢ V is a T -invariant subspace, and let PU denote orthogonal projection to U .
(c) ShowthatifU? isT-invariant,thenPUT =TPU. Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 7 of 10
Question 5 3 pts
Suppose that V is a nite-dimensional inner product space and let U ⇢ V be a subspace. Deneanisomorphism F :U0 !U?.
(You must show that the map you dene satises the required properties.)
Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 8 of 10
Question 6 4 pts
Let V be a complex inner product space and let T 2 L(V). SupposethatT 1 = 1 1 andT 2 = 2 2 for 1 , 2.
If T is normal, prove that h 1, 2i = 0.
Write your solution here
MATH1116 End of Semester Exam, 2019 — Book B, Page 9 of 10
Question 7 5 pts
Suppose that T : R2 ! R3 is a linear map. With respect to the standard bases for Rn ,
suppose that
261 0 137 MTT⇤ =60 1 07.
41015 Find T 26 1 37 .
Write your solution here
MT⇤T=261 037,and
MATH1116 End of Semester Exam, 2019 — Book B, Page 10 of 10