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Restate the last sentence in Theorem 2 using the concept of pivot columns: “If a linear system is consistent, then the solution is unique if and only if

If a linear system is consistent, then the solution is unique if and only if every column of thecoefficient matrix is a pivot column; otherwise, there are infinitely many solutions.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 2

Row Reduction and Echelon Forms

Introduction to Matrices

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

08:43

Suppose the coefficient ma…

02:44

a. Fill in the blank in th…

02:45

Use Theorem 3 (but not The…

02:05

Let $T : \mathbb{R}^{n} \r…

01:01

Determine whether the stat…

00:25

Determine whether each of …

00:26

The following theorem can …

01:00

00:24

09:31

Suppose a system of linear…

In this video, we have augmented matrix given by this matrix here, a B where a is a coefficient matrix to a linear system of equations in and different unknowns. If we like we can re express this matrix a B as follows, we can say a augmented with B would be equal to first calm. One of a column two of a All the way to With There's an unknowns will have and calms altogether, then augmented with the right hand side of the equations, which is the vector B. Let's hear assume that the system is consistent. Then what that tells us is the following There will be or, let's say, then there is no pivot in the far right. Call him in other words, that far right, calm here cannot contain a pivot. So this will mean that there exists because solution And what about uniqueness? We can also say that the solution is unique when, well, what should happen now for uniqueness? Well, if we have a unique solution, that means there is no free variables. But to have no free variables means every one of these columns here is a pivot column so we can say this solution is unique when all columns from the Matrix A. That's this portion or this portion here, not the augmented portion. So all comes from the Matrix, say our pivot columns. When all columns are pivot comes, then we have no free variables for a unique solution.

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