💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Stuck on your homework problem? This step-by-step video should help.

Try Numerade Free for 30 Days

Like

Report

Suppose $A$ is a $3 \times 3$ matrix and $b$ is a vector in $\mathbb{R}^{3}$ with the property that $A \mathbf{x}=\mathbf{b}$ has a unique solution. Explain why the columns of $A$ must span $\mathbb{R}^{3}$ .

If the equation $A x=b$ has a unique solution, then the associated system of equations does not have any free variables. If every variable is a basic variable, then each column of $A$ is a pivot column. So the reduced echelon form of A must be unitary matrix of $3^{*} 3 .$ A has a pivot position in each row. By Theorem 4 , the columns of A span R4.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 4

The Matrix Equation Ax D b

Introduction to Matrices

McMaster University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:18

Suppose $A$ is a $4 \times…

03:19

Let $A$ be a $3 \times 2$ …

04:16

Suppose $A$ is a $3 \times…

02:40

Construct a $3 \times 3$ m…

02:29

Suppose $A$ is an $m \time…

01:43

01:39

06:33

Suppose $A$ is an $n \time…

02:54

Construct a $3 \times 3$ n…

All right, so in problem 34 suppose A's of three by three matrix and A's B's a vector in our three with the property. That Exodus p D has a unique solution. And we want to explain why the column Self Bay must spend our three. So Okay, so in this case, since we know the leaner system they X equals B is has a unique solution. So that means if we consider the or commended matrix and furthermore, if we considered the rule reduced form off this matrix, it must has has to ruin form like this 81 1 a two to something on the diagonal, diagnose and have some stuff there some stuff here. But it doesn't matter. What are they? And we have zeros here, and I'm on the right hand side. We have B one b two B three as a constant. Okay, So this is the only impossible for this system to have a unique solution because only in this way we can solve a unique by the By the last row, we can solve the unique, unique x three, and furthermore that from the second road that X two will be determined by X three and x one would be determined by X two and x three. So this the only impossible way that we can have a unique solution for this system. So that means by observing this matrix, we have three people, two columns and three people. Columns means, uh, means axes. Um, yeah. So three people columns actually gives me, gives us the information that all the columns because A said three by three matrix And since every every single column is, is that people column So these columns are linearly independent, very independent, and furthermore, they must spend our three because in this case, we actually have three vectors and our three estimation three and those three Rector's Arlene Early independent so thes three directors must spend our free so

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 …

Suppose $A$ is a $4 \times 3$ matrix and $\mathbf{b}$ is a vector in $\mathb…

Let $A$ be a $3 \times 2$ matrix. Explain why the equation $A \mathbf{x}=\ma…

Suppose $A$ is a $3 \times n$ matrix whose columns span $\mathbb{R}^{3}$ . E…

Construct a $3 \times 3$ matrix $A,$ with nonzero entries, and a vector $\ma…

Suppose $A$ is an $m \times n$ matrix with the property that for all $\mathb…

Construct a $3 \times 3$ matrix, not in echelon form, whose columns span $\m…

Suppose $A$ is a $3 \times 3$ matrix and $y$ is a vector in $\mathbb{R}^{3}$…

Construct a $3 \times 3$ matrix, not in echelon form, whose columns do not s…

Suppose $A$ is an $n \times n$ matrix with the property that the equation $A…

Construct a $3 \times 3$ nonzero matrix $A$ such that the vector $\left[\beg…

05:55

Find an LU factorization of the matrices in Exercises $7-16$ (with $L$ unit …

02:30

If the columns of a $7 \times 7$ matrix $D$ are linearly independent, what c…

01:19

Unless otherwise specified, assume that all matrices in these exercises are …

08:29

Use partitioned matrices to prove by induction that for $n=2,3, \ldots,$ the…

02:57

In Exercises 29 and 30 , describe the possible echelon forms of the standard…

05:45

Find the inverses of the matrices in Exercises $29-32,$ if they exist. Use t…

06:39

Repeat the strategy of Exercise 33 to guess the inverse of $$A=\left…

09:33

In Exercises 11 and $12,$ determine if $\mathbf{b}$ is a linear combination …

03:15

In Exercises 13 and $14,$ determine if $\mathbf{b}$ is a linear combination …

04:30

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

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.