×

### Let's log you in.

or

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

×

or

## Math 340 - Week 7

by: Susan Ossareh

22

0

3

# Math 340 - Week 7 Math 340

Susan Ossareh
CSU

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

×
Unlock Preview

Covers the last few sections before midterm 1
COURSE
Intro-Ordinary Differen Equatn
PROF.
TYPE
Class Notes
PAGES
3
WORDS
CONCEPTS
Differential Equations
KARMA
25 ?

## Popular in Math

This 3 page Class Notes was uploaded by Susan Ossareh on Sunday March 6, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 22 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

×

## Reviews for Math 340 - Week 7

×

×

### What is Karma?

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

Date Created: 03/06/16
Math 340 Lecture – Introduction to Ordinary Differential Equations – February 29 , 2016 th What We Covered: 1. Course Content – Chapter 7: Matrix Algebra a. Section 7.5: Bases of a Subspace Continued i. Recap 1. Nullspace: ????????????????(????) ???????????? = ???? ∶ ???????? = 0 ???????????? ℝ ???? ???? 2. Definition: a subspace ???? ???? ℝ such that a. If ????,???? ???? ???? then x+y e V b. If xeV, ???? ???? ℝ then ???????? ???? ???? 3. Null(A) is a subspace of ℝ???? 4. Linear dependence and independence a. Definition: the vectors ???? ,…,???? ???? ℝ are linearly independent is 1 ???? ???? 1 1 ⋯+ ???? ???? =???? ???????????????????? ???????? ???? + ⋯+1???? = 0 ???? ii. Example: are ???? = (1,0) and ???? = (0,1) linearly independent? 1 2 ???????????????????????????? ????ℎ???????????? ???????????????????????? ???? 1???? 2ℝ ????????????ℎ ????ℎ???????? ???? ????1 1???? ???? 2 2 ???? 1,0 + ???? 0,1 = (0,0) 1 2 (????1,????2) = (0,0) ????1= 0 ???????????? ???? = 2 V1 and V2 are linearly independent because both a1 and a2 are equal ( ) iii. Example: are ???? 1 1,2 ???????????? ???? = (22,4) linearly independent? Suppose ????11,−2 + ???? −224 = 0.) (????1− 2???? 2−2???? + 1???? ) =2(0,0) ???? 1 2???? =20 − 2???? 1 4???? =20 1 −2 0 −2 4 0 1. Row Operation: 2R1+R2 1 −2 0 0 0 0 2. X2 is a free column so it’s an independent variable: ???? 2 ???? ????1− 2???? = 0 → ????1= 2???? ???? 1 = ????2 ????2 1 ???? = 2 ???? = 1 1 2 2 1,−2 + 1 −2,4 = (0,0) (2,−4 + −2,4 = (0,0) 3. They aren’t linearly independent, so they have to be linearly dependent iv. To find out if the vectors {x1, xk} are linearly independent, we find null(x), x=[x1, xk]. If null(x) is trivial then they are linearly independent, if not then they are dependent v. Bases of subspace 1. Definition: ???? = ???? ,???? ,…,???? ???????? ℝ ???? 1 2 ???? 2. B is a basis of V(subspace) if a. Span ???? ,…,???? )= ???? 1 ???? b. {????1,…,???? }????are linearly independent vi. Example: ???? = ℝ a basis for v is ???? = { 1,0 , 0,1 } 1. They are linearly independent 2. (????,????)????ℝ , then ????,???? = ???? 1,0 + ???? 0,1 ???? ???????????????? 1,0 ,(0,1) vii. Example: Find a basis for the null space of C 1 −1 0 2 0 ???? = 0 0 1 2 −1 1. The null space of C is all ????????ℝ 2. Cx=0, solve 1 1 0 2 0 0 0 0 1 2 −1 0 3. We can say x1 and x3 are pivot columns and the rest are free columns. Now we can back solve ????2= ????, ????4= 5, ???? =5???? ????3= ???? − 2???? ????1= ???? − 2???? X1 r-2s X2 R X3 = t-2s X4 S X5 t 4. V1, v2, v3 are linearly independent and null( c )= span (v1, v2, v3) so ???? = {????1,????2,????3} is a basis for null( c ) viii. In general, a basis for v is not unique but always has the same # of vectors. The number of vectors in B = dimensions of V b. Section 7.6: Square Matrices i. Definition: ????????????????− # ???????????????????????????? = # ???????????? ii. Matrix A is nonsingular if it’s a square matrix and Ax=b has a solution for any b. If not, its singular iii.Proposition: A is nonsingular if and only if A changes to a row echelon form without zeros in the diagonal iv. Proposition: If A is nonsingular then Ax=b has a unique solution v. Proposition: Ax=0 has only a trivial solution if and only if A is nonsingular. Hence the null(A)={0} Suggested Homework:  Study for exam 1  Section 7.5: 5, 10, 20, 22, 30, 38  Section 7.6: 8, 14, 20, 24 st Math 340 Lab – Introduction to Ordinary Differential Equations – March 1 , 2016 What We Covered: 1. Went over the practice exam in class 2. Course Content – Chapter 7: Matrix Algebra a. Section 7.6: Inverse of a Matrix i. Definition: ???????????????? is invertible if there exis????????????such that AB= 1 = BA we call ???? = −1 ???? the inverse of A ii. Proposition: A is invertible if A is nonsingular 3 1 iii.Example: Compute ???? −1 for A= −1 2 3 1 | 0 −1 2 0 1 1. Row reduce to the identity a. R1 +R2 = R2 3 1 1 0 0 7⁄ |1/3 1 3 b. 1/3R1 and 3/7R2 1 1/3 1/3 0 | 0 1 1/7 3/7 c. –R2/3 + R1=R1 1 0 2/7 −1/7 | 0 1 1/7 3/7 2. So the inverse of A is… ???? −1= 1 2 −1 7 1 3 3. In general… ???? ????????????= ???? ???? ???? ???? −1 1 ???? −???? ???? = ???????? − ???????? −???? ???? 4. This is only valid for 2x2 matrix Suggested Homework:  Study for exam 1  Section 7.6: 8, 14, 20, 24 Math 340 Lecture – Introduction to Ordinary Differential Equations – March 2nd, 2016 What We Covered: 1. More review for the exam a. Highlights i. Used the practice exam 2. Announcement a. No class on Friday!

×

×

### BOOM! Enjoy Your Free Notes!

×

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."

Janice Dongeun University of Washington

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

Jim McGreen Ohio University

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

### 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