Need help with your Discussion

Get a timely done, PLAGIARISM-FREE paper
from our highly-qualified writers!

glass
pen
clip
papers
heaphones

MATH 2001 US Linear Algebra Inner Product Space Equations Questions

MATH 2001 US Linear Algebra Inner Product Space Equations Questions

MATH 2001 US Linear Algebra Inner Product Space Equations Questions

Description

   

Unformatted Attachment Preview

School of Mathematics and Physics, UQ
MATH2001, Assignment 1, Summer Semester, 2022-2023
Due at 2:00pm 22 December. Each question marked out of 10. Total marks: 50
Submit your assignment online via the Assignment 1 submission link in Blackboard.
(1) Consider the equation

cos y + 2e?x cos x
y

y ? = 2e?x sin x ?
sin y
.
y
(a) Show that it is not exact, but it becomes exact when multiplying by the
integrating factor = yex .
?
(b) Solve the exact equation subject to the initial condition y(?) = . Present
2
your solution as a relation defining y implicitly as a function of x.
Show all working.
(2) Consider the non-homogeneous ODE
x2 y ?? ? 3xy ? + 4y = x2 ln x,
x > 0.
(a) Show that y1 = x2 and y2 = x2 ln x are solutions to the corresponding homogeneous ODE x2 y ?? ? 3xy ? + 4y = 0.
(b) Find the general solution of the non-homogeneous ODE.
Show all working.
(3) Consider the inner product space P2 (R) with inner product
p0 + p1 x + p2 x2 , q0 + q1 x + q2 x2 = p0 q0 + p1 q1 + p2 q2 , p0 , p1 , p2 , q0 , q1 , q2 ? R.
Let U = {1 + 2x + x2 }.
(a) Find U ? .
(b) Determine an orthogonal basis for U ? .
Show all working.
(4) Consider the vector space R4 endowed with the inner product
(u1 , u2 , u3 , u4 ) , (v1 , v2 , v3 , v4 ) = u1 v1 + 3u2 v2 + u3 v3 + 2u4 v4 .
Let U = span{u1 = (?2, 1, 0, 1), u2 = (0, 1, 2, 3)} be a subspace of R4 .
(a) Use Gram-Schmidt procedure to construct an orthonormal basis for U .
(b) Find the orthogonal projection of v = (?1, 2, 5, 1) in U and U ? .
Show all working.
(5) Let P2 (R) have the inner product,
Z 1
?p, q? =
p(x)q(x) dx,
?p, q ? P2 (R).
0
Find the best approximation of f (x) = x2 + x3 by polynomials in P2 (R). Show all
working.

Purchase answer to see full
attachment

Explanation & Answer:

5 Questions

User generated content is uploaded by users for the purposes of learning and should be used following our honor code & terms of service.

Have a similar assignment? "Place an order for your assignment and have exceptional work written by our team of experts, guaranteeing you A results."

Order Solution Now

Our Service Charter


1. Professional & Expert Writers: Eminence Papers only hires the best. Our writers are specially selected and recruited, after which they undergo further training to perfect their skills for specialization purposes. Moreover, our writers are holders of masters and Ph.D. degrees. They have impressive academic records, besides being native English speakers.

2. Top Quality Papers: Our customers are always guaranteed of papers that exceed their expectations. All our writers have +5 years of experience. This implies that all papers are written by individuals who are experts in their fields. In addition, the quality team reviews all the papers before sending them to the customers.

3. Plagiarism-Free Papers: All papers provided by Eminence Papers are written from scratch. Appropriate referencing and citation of key information are followed. Plagiarism checkers are used by the Quality assurance team and our editors just to double-check that there are no instances of plagiarism.

4. Timely Delivery: Time wasted is equivalent to a failed dedication and commitment. Eminence Papers are known for the timely delivery of any pending customer orders. Customers are well informed of the progress of their papers to ensure they keep track of what the writer is providing before the final draft is sent for grading.

5. Affordable Prices: Our prices are fairly structured to fit in all groups. Any customer willing to place their assignments with us can do so at very affordable prices. In addition, our customers enjoy regular discounts and bonuses.

6. 24/7 Customer Support: At Eminence Papers, we have put in place a team of experts who answer all customer inquiries promptly. The best part is the ever-availability of the team. Customers can make inquiries anytime.

We Can Write It for You! Enjoy 20% OFF on This Order. Use Code SAVE20

Stuck with your Assignment?

Enjoy 20% OFF Today
Use code SAVE20