Homogeneous linear equations with more variables than equations

Dependencies:

  1. Rank of a homogenous system of linear equations

Let $AX = 0$ be a system of $m$ linear equations in $n$ variables where $m < n$. Then there is a non-trivial solution.

Proof

$\operatorname{rank}(A) \le m < n \Rightarrow$ there is a non-trivial solution.

Dependency for:

  1. Linearly independent set is not bigger than a span

Info:

Transitive dependencies:

  1. /linear-algebra/matrices/gauss-jordan-algo
  2. /linear-algebra/vector-spaces/condition-for-subspace
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Field
  8. Vector Space
  9. Linear independence
  10. Span
  11. Integral Domain
  12. Comparing coefficients of a polynomial with disjoint variables
  13. Semiring
  14. Matrix
  15. Stacking
  16. System of linear equations
  17. Product of stacked matrices
  18. Matrix multiplication is associative
  19. Reduced Row Echelon Form (RREF)
  20. Elementary row operation
  21. Every elementary row operation has a unique inverse
  22. Row equivalence of matrices
  23. Matrices over a field form a vector space
  24. Row space
  25. Row equivalent matrices have the same row space
  26. RREF is unique
  27. Identity matrix
  28. Inverse of a matrix
  29. Inverse of product
  30. Elementary row operation is matrix pre-multiplication
  31. Row equivalence matrix
  32. Equations with row equivalent matrices have the same solution set
  33. Rank of a homogenous system of linear equations
  34. Rank of a matrix
  35. Basis of a vector space
  36. Linearly independent set is not bigger than a span