### Create a StudySoup account

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

Already have a StudySoup account? Login here

# 121 Class Note for MATH 220 at PSU

### View Full Document

## 18

## 0

## Popular in Course

## Popular in Department

This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 18 views.

## Similar to Course at Penn State

## Popular in Subject

## Reviews for 121 Class Note for MATH 220 at PSU

### 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: 02/06/15

Linear Combination and Span Section 44 0 De nitions from 44 1 Let 1117 1127 7 1 be vectors in a vector space V7 then any vector in the form 11111 12112 any where 117 127 7 an are scalars is called a linear combination of1117 1127 7 1 2 The vectors 1117 1127 7 1 in a vector space V7 are said to span V7 if every vector in V can be written as a linear combination of them That is7 for every 1 6 V7 there are scalars7 117 127 7 a such that va111a2112an11n 3 Let 1117 1127 7 11k be vectors in a vector space V7 the span of117 1127 7 11k is the set of all linear combinations of1117 1127 7 11k That is7 span117 1127 7 11k 1 1 11111 12112 unik where 117 127 7 an are arbitrary scalars 0 Theorem from 44 1 For any collection 1117 1127 7 wk of vectors in a vector space V7 the span of117 1127 7 11k is a subspace of V That is7 span 017 1127 7 11k is a subspace of V Linear Independence Section 45 0 De nitions from 45 1 Let 1117 1127 7 1 be vectors in a vector space V The vectors are said to be linearly dependent if there exists n scalars 017 027 7 077 not all zero such that 01111 0202 cnvn 0 If the vectors are not linearly dependent they are said to be linearly independent 2 Another form of the above de nition Let 1117 1127 7 1 be vectors in a vector space V These vectors are said to be linearly independent if 01111 02112 077177 0 1 implies that that 017 027 7 077 are all zero If equation 1 holds and one or more of the scalars is not zero7 then these vectors are said to be linearly dependent o Theorems from 45 1 Let A be an n x 71 matrix with columns 1117 1127 7 1 The columns of A for a linearly independent set if and only if the only solution to Ax 0 is z 0 2 Let A be an n x 71 matrix Then detA 31 0 if and only if the columns and rows of A are linearly independent Basis and Dimension Section 46 0 De nition from 46 1 A nite set of vectors 017 1127 7 077 is a basis for a vector space V if a 017 1127 7 077 is linearly independent b 017 1127 7 en spans V The dimension of a vector space V is the number of vectors in every basis for V The dimension of a vector space is denoted dim V If every basis has in nitely many vectors7 then V is said to be in nite dimensional7 and if the only vector in its basis is the zero vector then dim V 0 o Theorems from 46 1 117 127 lf117 1127 7 077 is a basis for a vector space V7 then any vector in V can be represented as a unique linear combination of the vectors in the basis 7 U77 and U17 u27 7 um are bases for V then m n This says that the number of terms in all bases for a vector space is the same If H is a subspace of a vector space V7 then dim H S dim V

### 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

#### "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!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "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."

#### "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."

### Refund Policy

#### STUDYSOUP CANCELLATION 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 support@studysoup.com

#### STUDYSOUP REFUND POLICY

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: support@studysoup.com

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 support@studysoup.com

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.