Proof on homogeneous equations

  • Context: Undergrad 
  • Thread starter Thread starter toxi
  • Start date Start date
  • Tags Tags
    Homogeneous Proof
Click For Summary
SUMMARY

The discussion centers on proving that if M > N, any system of N homogeneous equations in M unknowns has infinitely many solutions. Participants emphasize the use of an MxN matrix to demonstrate this concept, highlighting that the determinant will indicate linear dependence among the vectors spanning the column space. The conclusion is that the remaining columns, after accounting for leading 1's, can take on arbitrary values, confirming the existence of multiple solutions.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically homogeneous equations
  • Familiarity with matrix representation and determinants
  • Knowledge of vector spaces and linear dependence
  • Basic skills in solving systems of equations
NEXT STEPS
  • Study the properties of homogeneous systems of equations
  • Learn about the rank-nullity theorem in linear algebra
  • Explore the concept of linear independence and dependence in vector spaces
  • Investigate the implications of determinants in matrix theory
USEFUL FOR

Students of linear algebra, mathematicians, and educators seeking to deepen their understanding of homogeneous equations and their solutions.

toxi
Messages
12
Reaction score
0
I need some help here... I've got the following assignment to do

Prove that if M>N then any system of N homogeneous equations in M unknowns has many solutions.

I am a bit stuck with this one. I thought about creating a MxN Matrix and to display the determinant with 1's.

and then say about the remaining colums after the rows with leading 1's stop (r = M-N), that they can represented by any value so there are many solutions
is that correct?
 
Physics news on Phys.org
i don't know what you're saying but yes if you write down a matrix with M columns and N rows you'll see that you have a linearly dependent set of vector spanning the column space hence many solutions.
 

Similar threads

Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K
High School Is it invalid to redefine the sgn function in this way in a proof?
  • · Replies 15 ·
Replies
15
Views
5K
High School Understanding Invertible Matrices and Homogenous Systems
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Dimension and solution to matrix-vector product
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Solving matrix commutator equations?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Proof concerning the Four Fundamental Spaces
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Trouble with a passage in Zariski & Samuel's Commutative algebra text
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Infinite solutions to homogeneous system?
  • · Replies 2 ·
Replies
2
Views
6K