New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

Linear Algebra (Honors)

by: Kaylin Wehner

Linear Algebra (Honors) MATH 115AH

Kaylin Wehner
GPA 3.55

C. Manolescu

Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

C. Manolescu
Class Notes
25 ?




Popular in Course

Popular in Mathematics (M)

This 4 page Class Notes was uploaded by Kaylin Wehner on Friday September 4, 2015. The Class Notes belongs to MATH 115AH at University of California - Los Angeles taught by C. Manolescu in Fall. Since its upload, it has received 98 views. For similar materials see /class/177807/math-115ah-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.


Reviews for Linear Algebra (Honors)


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/04/15
Math 115 AH Practice Midterm 2 Linear Algebra You have 50 minutes No books and notes are allowed 1 TrueFalse Circle the correct answer No justi cations are needed in this exercise 1 point each In the questions below7 V is a vector space over a eld IF 1 A linear transformation T V a V is invertible if and only if T is injective T 2 Assume A B E Mnm are invertible Then AB is invertible and AB 1 A lB l F We actually have AB 1 B lA l 3 Assume T V a V is invertible7 and BC are ordered bases for V Then T 1 T71 We actually have T 1 TA F 4 If A is a matrix7 then rankA rankAt T 5 If A E M3X3R satis es At 7A then A is not invertible T Use determinants 6 Similar matrices always have the same eigenvalues T 7 If A is an eigenvalue of an operator T V a V then each vector in EA is an eigenvector for T F Zero is not an eigenvector7 but lies in EA 8 If W1 W2 W3 Q V are subspaces such that V W1W2W3 and W1 Wg Wg 0 then V W1 69 W2 69 W3 Consider W1 W2 Xy plane and W3 Z aXis in R3 9 If V is an inner product space and 1 E V is such that ltvwgt 0 for all w E V then 0 1 T 10 If V is an inner product space and S Q V is a subset7 then the orthogonal complement of S is a subspace of V T 2 10 points Determine whether the matrix A lt35 31 E M2X2R is diagonaliz able The characteristic polynomial is t2 7 tr At det A t2 9 which does not split over R it has no roots Hence A is not diagonalizable 3 10 points Let V EUR be the real vector space of real polynomials of degree at most one Consider the linear functionals fhfg E V given by up 1plttgtda f2p p 0 Show that f1 and f2 form a basis for V For a polynomial pt at b we have f110 02 t b and f210 a We prove that f1 and f2 are linearly independent Indeed7 suppose there exist 041 042 E R with a1f1 cvng 0 Then a 041E b 042a 0 for all ab E R Choosing a 0b 1 we get 041 0 Choosing a Lb 0 we get 042 0 Thus f1 and f2 are linearly independent Since V is two dimensional7 so is V Two linearly independent vectors in a two dimensional space must form a basis MATH 115A MIDTERM SOLUTION Monday7 February 9th 2009 Exercise 1 10 points Bases Let F be a eld Find bases for the following subspaces of F5 justify your answer W1 01702703704705 6 F5 3 a2 as i 04 0 W2 a17a27a37a47a5 E F5 a1 13 a4 and a2 a5 0 What are the dimensions of W1 and W2 For W1 it clear that the map ltlgt F5 7 F de ned by 19017 12703704705 a2 as i 04 is a linear transformation and Nltlgt W1 Applying the dimension theorem7 it follows that dim W1 4 and a basis for W1 is 0 0 0 1 7 71 0 OHHMO l 0 0 51 0 7 07 0 0 0 l A vector x 6 W2 is necessarily of the form x 1l117 7b So dim W2 2 and a basis for W2 is OOHO 1 0 6217 1 0 l H Exercise 2 10 points Lagrange Polynomials Let F be a eld and n E N be an integer Denote by PnF the vector space of all polynomials P with coef cients in F such that degP S n The indeterminate will be denoted by t Let zo an E F be n1 distinct scalars For k E 07 7n7 de ne the polynomial 1t 7 960 1t 7 mm 7 n1 1t 7 96a 96k 7 760 96k 7 mm 7 n1 9 7 ml 1 LIN 1 Compute Lkzj for all j k E 0 n Then show that the subset L0 L1 Ln is a basis for A straightforward computation shows that Lkzj jk ie Lkxj1ifj k and 0 ifj 51 k Let now a0a1an be 71 1 scalars and assume that aOLO alLl anLn 0 the zero polynomial This implies in particular that for any j 0 0010 11111 anLnj 01le 07quot Consequently a0 a1 an 0 and the subset B L0L1Ln is linearly independent Since dim PnF n 1 it follows that B is indeed a basis for Let y0y1 yn E F be any scalars Use the previous question to show that there exists one and only one polynomial P E PnF such that Pzk yk for all k E 0 n Assume that such a polynomial P exists We may decompose P using the basis 6 Then there are scalars a0a1 an such that P Z aij j0 Necessarily yj Px ajLJxj 17 So P is necessarily unique and P does exist It suf ces to set P 2 ijj j0 D Exercise 3 10 points Applying the dimension theorem Let V be a nite dimensional vector space and let T V gt V be a linear transformation We assume that nullityT nullityT2 1 Using the assumption together with the dimension theorem prove that NT NT2 and RT RT2 The inclusion NT C NT2 is obvious Since dimNT dim NT2 it follows that NT NT2 Applying the dimen sion theorem twice for T and T2 we get dimV nullityT rankT nullityT2 rankT2


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Kyle Maynard Purdue

"When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the I made $280 on my first study guide!"

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.