Linear Algebra uniqueness of solution

Click For Summary

Homework Help Overview

The discussion revolves around the uniqueness of solutions in linear algebra, particularly in the context of a system represented by an augmented matrix. Participants explore the implications of having free variables and leading entries in the matrix.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the consistency of the system based on the structure of the matrix, questioning the nature of free variables and leading entries. There are inquiries about the notation used in the matrix representation and how it translates to a system of equations.

Discussion Status

The discussion is active, with participants providing clarifications on notation and the definitions of leading entries. Some express agreement with the initial interpretation of the problem, while others raise concerns about potential contradictions in the matrix entries that could affect the existence of solutions.

Contextual Notes

There is a focus on the definitions of leading entries and free variables, with some participants noting the importance of context in understanding the implications of the matrix form. The discussion also highlights the need for clarity regarding the values represented by the symbols in the matrix.

Sunwoo Bae
Messages
60
Reaction score
4
Homework Statement
determine if the system is consistent. If the system is consistent, determine if the solution is unique
Relevant Equations
none
1619352138083.png


My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution).
As the first column is all 0s, x1 would be a free variable.
Because the system with free variable have infinite solution, the solution is not unique.
In this way, the matrix is consistent, and has no unique solutions.

Would this be a correct answer, and is the explanation suitable?

Thank you!
 
Physics news on Phys.org
Can you explain the notation a little bit more here? are all the *s intended to be the same number? Are all the little boxes intended to be the same number?

How do you get from a matrix to a system of equations? Is the matrix written as [A|b] for the system of equations Ax=b?
 
Office_Shredder said:
Can you explain the notation a little bit more here? are all the *s intended to be the same number? Are all the little boxes intended to be the same number?

How do you get from a matrix to a system of equations? Is the matrix written as [A|b] for the system of equations Ax=b?
The square represents leading entries (they do not have to be the same number). the star represents any numbers (the stars do not have to be the same numbers). 0 is the number 0.

The matrix above is the augmented matrix.

Sorry for the lack of explanation in the question...
 
I think you are entirely correct
 
Sunwoo Bae said:
The square represents leading entries (they do not have to be the same number). the star represents any numbers (the stars do not have to be the same numbers). 0 is the number 0.

The matrix above is the augmented matrix.

Sorry for the lack of explanation in the question...
Is there more to say about the entries? Are all the stars and squares non-zero? It seems like it would be easy to make the first two rows contradictory and therefore no solution.
 
I think the squares are intended to be non zero (that's the definition of a leading entry) but the stars could be zero.

I also agree with the answer, as long as the context is obvious that you are talking about a row reduced echelon form of a matrix (if not, you might want to say something about it, since the lack of a zero row is not sufficient)
 
  • Like
Likes   Reactions: FactChecker
Office_Shredder said:
I think the squares are intended to be non zero (that's the definition of a leading entry)
Good point. I missed that. Thanks.
 

Similar threads

Conflicting definitions of linear independence
  • · Replies 30 ·
2
Replies
30
Views
2K
Determining value of r that makes the matrix linearly dependent
  • · Replies 4 ·
Replies
4
Views
2K
Show that the system of equations has no unique solution
  • · Replies 2 ·
Replies
2
Views
1K
Solution Sets of Linear Systems
  • · Replies 14 ·
Replies
14
Views
3K
Intro to Linear Algebra - Nullspace of Rank 1 Matrix
  • · Replies 8 ·
Replies
8
Views
2K
Given specific v, dimension of subspace of L(V, W) where Tv=0?
  • · Replies 15 ·
Replies
15
Views
2K
System of equations and solving for an unknown
  • · Replies 7 ·
Replies
7
Views
2K
Linear algebra unique solutions
  • · Replies 1 ·
Replies
1
Views
2K
Dot diagrams and Jordan canonical forms
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K