Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Like

Report

In Exercises $7-12$ , describe all solutions of $A \mathbf{x}=0$ in parametric vector form, where $A$ is row equivalent to the given matrix. $$\left[\begin{array}{rrr}{3} & {-9} & {6} \\ {-1} & {3} & {-2}\end{array}\right]$$

$X=x_{2}\left[\begin{array}{l}{3} \\ {1} \\ {0}\end{array}\right]+x_{3}\left[\begin{array}{c}{-2} \\ {0} \\ {1}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

Campbell University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

03:23

In Exercises $7-12$ , desc…

04:08

04:06

07:25

07:52

04:53

In Exercises 5 and $6,$ fo…

00:42

In Exercises $9-12$, the r…

00:28

00:36

00:24

in this video, we have a matrix equation given to be a X equals zero. So it's a homogeneous system of equations. And we're also told that a ISRO equivalent to the matrix we see displayed here well, using that information, the idea that we would like to pursue is what is the entire solution set to this system of equations? Well, to do that, we're going to have to roll, reduce. Let's start back on this matrix A that we see displayed. And I'm going to make my first role operation to divide Row one by three that obtains one naked of three and two and below. We have negative 13 and negative, too. We noticed. Now that Row two is suspicious. It's just row one multiplied by negative one. And when that happens, we can immediately say that this is row equivalent to one negative 32 with a zero down below. Or in other words, if the rose are a non constant multiple of each other, one of them is eliminated. So now that we have a in its role reduced echelon form, we're ready to go back and consider the solution set to this system of equations indicated here. We have to be careful, though, because our Matrix say has not been augmented. So when we go to Row One, no one is thick. Waysh in x one minus three x two plus two x three equals zero. Row two has no new information, so we'll just record that x two is free and x three is free. These air free variables since the Onley pivot column is column one corresponding to variable x one, making both x two and X three free variables. Now we're ready to describe our solution set to this system by a step by step process you can write to. The X is equal to x one x two x three So this is our set up and we'll aren't going to do the following first go to the equation for X one and solve it for X one. We would be forced to add three x two double sides, so that would go here minus two x three. So now if you read this factor equation, we're saying X one equals three x two minus two x three And that's no different from this equation displayed here. Now the trick is what to do with the free variables X two and x three. Well, there's no restriction on next to since it's free, so we can say X two equals x two. But the vector equation we have here we would just write in an X two in this place. Likewise, X three is free means there's no restriction so we can write the trivial equation. X three equals x three and that means X three gets placed here in that way. Now, with this set up, we're ready to begin describing the Parametric Vector form. Now. One trick in the vector form is the free variables are going to be equal to the scale. Er's in this linear combination. Let me start out by writing x two and a plus x three here. Now we just need to determine what the vectors that these scale er's X two x three would be multiplying well for the first vector. It's associated with the X two variable, and we just pull the coefficients, which are 31 and 00 because there is no value here or if you prefer, you could have called it zero times x two just for the sake of organization. Now for the X three variable, we go to the x three column here and take negative too. Zero and one here X two and x three are parameters or free variables. And although those things are just different ways of saying the X two x three are in our so any real number with no restrictions. So this is the parametric vector form for this system of equations.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

In Exercises $7-12$ , describe all solutions of $A \mathbf{x}=0$ in parametr…

In Exercises 5 and $6,$ follow the method of Examples 1 and 2 to write the s…

In Exercises $9-12$, the reduced row echelon form of the augmented matrix of…

02:01

$[\mathbf{M}]$ In Exercises $37-40,$ let $T$ be the linear transformation wh…

02:24

Find the area of the polygon with the given vertices.$\mathrm{L}(-3,1), …

01:38

In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the s…

03:16

REPEATED REASONING $\overline{A B}$ is divided into four congruent segments,…

05:20

A steam plant burns two types of coal: anthracite (A) and bituminous (B). Fo…

00:54

01:55

Why is the question "Is the linear transformation $T$ onto?" an ex…

00:40

Copy and complete the tables for the regular polygons shown. In these tables…

02:02

The given matrix determines a linear transformation $T .$ Find all $\mathbf{…

03:05

Consider the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3},$ and $…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.