### Create a StudySoup account

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

Already have a StudySoup account? Login here

# MATH-M303 Section 1.2 Notes MATH-M 303

IU

GPA 4.0

### View Full Document

## About this Document

## 5

## 0

## Popular in Linear Algebra for Undergraduate

## Popular in Mathematics

This 5 page Class Notes was uploaded by Kathryn Brinser on Friday September 9, 2016. The Class Notes belongs to MATH-M 303 at Indiana University taught by Keenan Kidwell in Summer 2016. Since its upload, it has received 5 views. For similar materials see Linear Algebra for Undergraduate in Mathematics at Indiana University.

## Reviews for MATH-M303 Section 1.2 Notes

### 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: 09/09/16

M303 Section 1.2 Notes- Row Reduction and Echelon Forms 8-26-16 Elementary row operations- to obtain equivalent systems o Row replacement- replace some row ???? with row ???? + ????(row ????′) for some ???? ???? ℝ; adding a multiple of another row to the row in questio????; ???? ???? ???? + ???????? ′ o Row swap- ????????↔ ???? ???? o Scaling- multiply row by nonzero number ????; ???? →???????????? o DON’T FORGET to do operations to augmented column too o Every operation reversible o 2 matrices ???? and ???? are row equivalent (???? ~ ????) if ???? can be obtained from ???? via finite sequence of elementary row operations If ???? ~ ????, then ???? ~ ???? and ???? ~ ???? iff ???? ~ ????; relation of row equivalency is symmetric ′ ′ o If we perform a row op on any augmented matrix ???? ???? to get ???? ???? , any solution to original system is a solution of system corresponding to ???? ???? Since row ops reversible, converse true; systems corresponding to ????|???? and ???? ???? are equivalent o Typically put tilde (~) somewhere near expressions of row ops to show that resulting matrix is similar to previous step Ex. Informal row reduction; find all solutions (if any) to the following system: ????1− 2???? 2 ???? =30 2???? 2 8???? =38 5???? − 5???? = 10 1 3 1 −2 1 0 o Let the system be represented by augmented matrix ???? ???? =0[ 2 −8 8 ] 5 0 −5 10 o Make zeroes below diagonal entries 1 −2 1 0 ???? → ???? − 5???? ~ [ 0 2 −8 |8 ] 3 3 1 0 10 −10 10 1 1 −2 1 0 ????2→ ????2~ 2 [0 1 −4 |4 ] 0 10 −10 10 1 −2 1 0 ????3→ ???? 3 10???? ~ 2 0 1 −4 | 4 ] 0 0 30 −30 1 −2 1 0 1 ????3→ 30???? 3 [0 1 −4 4 ] 0 0 1 −1 o Can see 3 = −1, but still need substitution; can continue to simplify by making zeroes above diagonal 1 −2 0 1 ????1→ ???? 1 ???? ~3 [0 1 −4 4 ] 0 0 1 −1 1 −2 0 1 ????2→ ???? 2 4???? ~ 3 0 1 0 0 ] 0 0 1 −1 1 0 0 1 ???? → ???? + 2???? ~ [ 0 1 0 | 0 ] 1 1 2 0 0 1 −1 o Simplified system shows solutio1: ???? = 2,???? = 03???? = −1 (1,0,−1) Existence and Uniqueness of Solutions o Is system consistent? If yes, is solution unique? o Ex. Inconsistent system: 0 1 −4 8 [???? ???? = [2 −3 2 1 ] 4 −8 12 1 Want nonzero in11 : 2 −3 2 1 ???? 1 ???? ~2 0 1 −4 |8] 4 −8 12 1 Want 0 in bottom left: 2 −3 2 1 ???? 3 ???? −32???? ~1[ 0 1 −4 8 ] 0 −2 8 −1 Want -2 in3???? to be 0: 2 −3 2 1 ???? 3 ???? +32???? ~2[ 0 1 −4 8 ] 0 0 0 15 Notice that i3 ???? , 0 ≠ 15, so no solution can satisfy equation 3 Will always get 0 = nonzero ???? somewhere when row reducing if system inconsistent Nonzero row/column- at least one nonzero entry Leading entry- leftmost nonzero entry in a nonzero row, counting augmented column Echelon form (EF)- every matrix can be converted using finitely many elementary row ops to one in EF: o Must satisfy 3 conditions: All nonzero rows are above any zero rows; row swap zero rows to bottom if needed Each leading entry is in a column to right of any nonzero entry above it Each entry directly below leading entry must is 0 (implied by condition 2) o Infinitely many solutions and ways to get to EF o Ex. Matrix in EF: ???? 1 3 [0 ???? 4] leading entries bolded 0 0 0 Reduced Echelon form (REF)- using more row ops, matrix is in REF in addition to being in EF if: o All leading entries are 1’s (known as leading 1’s) o All entries above leading entries are zeros o Only one solution, many ways to get to REF from EF o For matrix ????, write ????????????(????) for unique matrix in REF row equivalent to ???? o Ex. Matrix in REF: ???? 0 3 [0 ???? 4] same as previous matrix, except 1 a2ove ???? ’s leading 1 is now 0 0 0 0 o If matrix ???? in EF, doing further row ops without breaking EF will not cause positions of leading entries to change, though entries themselves may change ( ) ( ) Pivot position in matrix ????- location corresponding to a leading 1 in ???????????? ???? or a leading entry in ???????? ???? o Pivot column- one that contains pivot position o Pivot positions/columns of ???? can be determined from any EF and tell us answers to existence and uniqueness questions for solutions to corresponding system o Ex. Row reduce the given matrix ???? to Echelon form and locate its pivot positions. 0 −3 −6 4 9 ???? = [−1 −2 −1 3 1 ] −2 −3 0 3 −1 1 4 5 −9 −7 ???? 4 5 −9 −7 −1 −2 −1 3 1 ????1⟷ ???? ~4 −2 −3 0 3 −1 ] gives 1 leading 1 as pivot position 0 −3 −6 4 9 1 4 5 −9 −7 ????2→ ???? 2 ???? 1nd ???? 3 ???? +32???? ~1 [0 ???? 4 −6 −6 ] 0 5 10 −15 −15 0 −3 −6 4 9 Pivot column 1 complete; now move on to pi2ot 2 in ???? 1 4 5 −9 −7 1 1 0 1 2 −3 −3 ????2→ ????2a2d ???? →3???? ~5 3 [0 1 2 −3 −3 ] 4 9 0 −3 16 4 5 −9 −7 ????3→ ???? 3 ???? 2nd ???? 4 ???? +43???? ~2 [0 1 2 −3 −3 ] 0 0 0 0 0 0 0 0 −5 0 1 4 5 −9 −7 ???? ⟷ ???? ~ [ 1 2 −3 −3 ] 3 4 0 0 0 −5 0 0 0 0 0 0 Matrix now in EF; pivot positions: ???? 4 5 −9 −7 [0 ???? 2 −3 −3 ] 0 0 0 −???? 0 0 0 0 0 0 Pivot columns- 1, 2, 4 (L to R) Formalizing Row Reduction 0 3 −6 6 4 −5 o Procedure to arrive at algorithm shown u3in−7???? =8[ −5 8 9 ] 3 −9 12 −9 6 15 o Steps 1-4 known as forward phase; step 5 produces unique result, known as backward phase o Step 1: Start in leftmost nonzero column (will be a pivot column); pivot position should be at top o Step 2: Row swap if necessary to put a nonzero entry in first pivot position 3 −9 12 −9 6 15 ????1⟷ ???? ~3 3 −7 8 −5 8 9] 0 3 −6 6 4 5 Can aquire 1 as pivot by dividing by 3: ???? −3 4 −3 2 5 ???? → ???? ~ 3 −7 8 −5 8 9 ] 1 3 1 0 3 −6 6 4 5 o Step 3: Make zeros below current pivot using row replacement 1 −3 4 −3 2 5 ????2→ ???? 2 3???? 1 [0 2 −4 4 2 −6 ] 0 3 −6 6 4 5 o Step 4: Cover/ignore current pivot row and all rows above it; apply steps 1-3 to remaining submatrix and repeat until there are no nonzero rows to modify 1 −3 4 −3 2 5 Ignore 1 , move to pivo2 [0 ???????? −4 4 2 −6 ] 0 3 −6 6 4 5 1 −3 4 −3 2 5 ????2→ ???? ~2 0 ???? −2 2 1 3 ] 2 0 3 −6 6 4 5 1 −3 4 −3 2 5 ???? → ???? − 3???? ~ 3 3 2 [0 ???? −2 2 1 3 ] 0 0 0 0 1 4 1 −3 4 −3 2 5 Ignore 2 , move to pivot 3n[0 1 −2 2 1 3] 0 0 0 0 ???? 4 Nothing to do below pivot 1; guaranteed to be in EF at this point o Step 5: To get from EF to REF, start with rightmost pivot; moving up and to left, make zeros above all pivot entries and scale any pivots to 1 if necessary 1 −3 4 −3 0 −3 ????1→ ???? 1 3???? a2d ???? → 2 − ????2~ 3 [0 1 −2 2 0 −7 ] 0 0 0 0 ???? 4 1 0 −2 3 0 −24 ????1→ ???? 1 3???? ~2 [0 1 −2 2 0 −7 ] now in REF 0 0 0 0 1 4 Extracting Solutions to Systems from REF o Once an augmented matrix of a system in REF, we can write out simplified equation for resulting equivalent system and easily find all solutions o Ex. Suppose a system has the following reduced augmented matrix: 1 0 −5 1 [ | ] 0 1 4 4 0 0 0 0 Column 1 represents coefficient1of ???? in each equation, column 2 coef2icient of ???? , etc., augmented column represents constants that equations equal Can be rewritten: ????1+ 0????2− 5???? 3 1 0????1+ ????2+ 4???? 3 4 0????1+ 0????2+ 0???? 3 0 ???? ,???? are pivot variables/basic variables- correspond to pivot columns of augmented matrix 1 2 ????3is free variable- can be anything; its column has no pivot, and each equation can be satisfied by picking random values f3r ???? Appear when matrix has more columns than rows- some must lack pivot entries o Augmented column can be pivot column or not- if yes, system has no solution Gives row of form 0 0 … 0 | ???? , which is impossible o Once augmented matrix in REF, we can write parametric description of solution set Express each pivot variable in terms of any free variable Ex. Find the solution set for the following system (if any) and write it in parametric form. 1 6 2 −5 −2 −4 0 0 2 −8 −1 | 3 ] in EF only 0 0 0 0 1 7 ???? 6 0 3 0 0 REF ends up a0 [ 0 ???? −4 0 | 5 0 0 0 0 ???? 7 Pivot variable1: 3 5???? ,???? Remaining must be free2 4 ,???? Parametric description of solution set: ????1= −6???? −23???? 4 ????2= ???????????????? ????3= 4???? 4 5 ????4= ???????????????? { ????5= 7 Actual (specific) solutions given after choosing values for free variables More on Existence and Uniqueness o In general, can answer questions about existence and uniqueness from EF Do not need REF- EF will show presence or lack of “bad rows” of form 0,0,…0 = ???? where ???? ≠ 0 Only way system inconsistent is presence of bad row(s) o Ex. Determine whether the system with the following augmented matrix is consistent. 0 3 −6 6 4 −5 [???? ???? = [3 −7 8 −5 8 | 9] 3 −9 12 −9 6 15 3 −9 12 −9 6 15 Arrive at [0: 1 −2 2 1 |−3 ] 0 0 0 0 1 4 No bad row- consistent Has infinitely many solutions- free variables present o Theorem- A system is consistent iff the augmented column of its augmented matrix is not a pivot column (ie. any EF of matrix does not have row of form 0 0 … 0 ????] where ???? ≠ 0 If system consistent, then system has unique solution iff no free variables present (ie. every column except augmented has a pivot) o No free variable = unique solution o Free variable + consistent = infinite solutions o Free variable + inconsistent = no solution Should always go to REF to solve systems UNLESS: o EF shows no solution/inconsistency o EF shows unique solution- use back substitution o Essentially- go to REF if free variables present

### 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

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

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

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "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.