Instructor(s): Joan and Griff Second Semester, 2021 Mathematical Sciences Institute
Australian National University
MATH1116, Advanced Mathematics and Applications 2 Assignment 9
Due via WebAssign and Gradescope by Sunday 17th October.
• The WebAssign portion of this assignment can be submitted a day late up until the Monday evening, without any extensions allowable beyond that point (because solutions become available on WebAssign immediately at the end of Monday).
• No submissions of the ‘Show Working’ portion of assignments will be accepted after the release of solutions.
• A reminder that you have agreed to Academic Integrity stipulations, as per the Academic Integrity Agreement that is linked to from the Week 2 block on Wattle and previous information provided in the Assignment 1 PDF.
WebAssign
See the WebAssign platform for the questions (worth 32 marks out of 50 for this assign- ment). Read the instructions at the top of the quiz for further details.
To login to WebAssign you should go to https://www.webassign.net/wa-auth/login and use your university email address as your username, with the password that you have previously set (not your main ANU password).
If you need to set up your password or have forgotten it, you can use the password reset process from the login page.
The ‘Show Working’ questions for this assignment, to be submitted via Gradescope, appear on the next page.
MATH1116, Assignment 9 2
Analysis (with some Linear Algbera)
Question 1. Differentiability (Show Working) 9 points
Suppose Rn is made into a real inner product space by giving it the standard dot product as its inner product ⟨·, ·⟩, such that the standard Euclidean norm is the norm.
Let T be a self-adjoint linear operator on Rn, and suppose p ∈ Rn. Define a function f : Rn −→ R by
f(x) = ⟨Tx,x⟩. Show that f is differentiable at p, with
dfp(v) = 2⟨Tp,v⟩
Note that you may assume the result, from the workshops, that T must be continuous.
Linear Algebra
Question 2. Orthogonal Projection (Show Working) 9 points
(Continuation from previous assignment.)
Suppose that V is a finite dimensional vector space and let P ∈ L(V ) satisfy P 2 = P .
In Assignment 8, we showed that you can define an inner product on V so that P is or- thogonal projection to some subspace.
This week’s question: If V is already equipped with an inner product, what further con- dition on P is required to show that it is orthogonal projection with respect to the given inner product? Prove your answer.
for all v ∈ Rn.