True or False: A Linear System Must Have a Unique Solution

  • Context: Undergrad 
  • Thread starter Thread starter Valour549
  • Start date Start date
Click For Summary
SUMMARY

A linear system with homogeneous equations does not necessarily have a unique solution; the statement that it must is false. The trivial solution exists, but there can also be infinitely many solutions depending on the system's parameters. This confusion arises from the interpretation of linear systems in the context of Elementary Linear Algebra, specifically in Anton's textbook. The assertion in the book that a homogeneous linear system has a unique solution is incorrect.

PREREQUISITES
  • Understanding of linear systems and their properties
  • Familiarity with homogeneous equations in linear algebra
  • Knowledge of solution types: trivial and non-trivial solutions
  • Basic concepts of vector spaces and dimensions
NEXT STEPS
  • Study the properties of homogeneous linear systems in depth
  • Learn about the role of the rank of a matrix in determining solution uniqueness
  • Explore the concept of vector spaces and their dimensions
  • Review examples of linear systems with infinite solutions
USEFUL FOR

Students of linear algebra, educators teaching the subject, and anyone seeking to clarify concepts related to linear systems and their solutions.

Valour549
Messages
57
Reaction score
4
So I'm reading chapter 1.1 of Elementary Linear Algebra by Anton and there's this true-false question:

"A linear system whose equations are all homogenous must have a unique solution."

Taking this question simply as it is I think the answer should be false, because while there's always the trivial solution, it's also possible for there to be infinite solutions. Yet the book gives the answer as True? I think it's pretty confusing and doesn't give the student confidence when the very first question has the wrong answer.
 
Physics news on Phys.org
The book is wrong.
 

Similar threads

Undergrad A true or false statement concerning condition number of a matrix
  • · Replies 2 ·
Replies
2
Views
2K
High School What unique contributions to math did linear algebra make?
  • · Replies 19 ·
Replies
19
Views
4K
High School System of linear equations: When are we guaranteed a unique solution?
  • · Replies 5 ·
Replies
5
Views
2K
High School Attempt to solve this system of three linear equations
  • · Replies 10 ·
Replies
10
Views
3K
MHB Verifying Answers to True/False Questions about Matrices
  • · Replies 3 ·
Replies
3
Views
2K
MHB How Can We Construct a Unique Linear System From Given Conditions?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Different norms and how L1 leads to the sparsest solution
  • · Replies 1 ·
Replies
1
Views
2K
Is this statement true or false?
  • · Replies 6 ·
Replies
6
Views
2K
MHB Solve System of Equations: Unique Solution & Geometric Explanation
  • · Replies 1 ·
Replies
1
Views
2K
MHB Finding Unique solution for system of linear equations
  • · Replies 1 ·
Replies
1
Views
2K