Equations with row equivalent matrices have the same solution set

Dependencies:

  1. Row equivalence of matrices
  2. System of linear equations
  3. Row equivalence matrix
  4. Product of stacked matrices
  5. Matrix multiplication is associative

Let $AX = B$ and $CX = D$ be systems of $m$ linear equations in $n$ variables. Then if $[A|B]$ is row equivalent to $[C|D]$, then $AX = B$ and $CX = D$ have the same solution set.

Proof

Since $[A|B]$ is row equivalent to $[C|D]$, there is an invertible matrix $R$ such that $R[A|B] = [C|D]$.

\[ [C|D] = R[A|B] = [RA|RB] \Rightarrow C = RA \wedge D = RB \]

\[ AX = B \Rightarrow R(AX) = RB \Rightarrow (RA)X = RB \Rightarrow CX = D \] Therefore, a solution to $AX = B$ is also a solution to $CX = D$. \[ CX = D \Rightarrow R^{-1}(CX) = R^{-1}D \Rightarrow (R^{-1}C)X = R^{-1}D \Rightarrow AX = B \] Therefore, a solution to $CX = D$ is also a solution to $AX = B$.

Therefore, $AX = B$ has the same solution set as $CX = D$.

Dependency for:

  1. Rank of a homogenous system of linear equations

Info:

Transitive dependencies:

  1. /sets-and-relations/equivalence-relation
  2. Group
  3. Ring
  4. Polynomial
  5. Integral Domain
  6. Comparing coefficients of a polynomial with disjoint variables
  7. Field
  8. Semiring
  9. Matrix
  10. Stacking
  11. System of linear equations
  12. Product of stacked matrices
  13. Matrix multiplication is associative
  14. Elementary row operation
  15. Every elementary row operation has a unique inverse
  16. Row equivalence of matrices
  17. Identity matrix
  18. Inverse of a matrix
  19. Inverse of product
  20. Elementary row operation is matrix pre-multiplication
  21. Row equivalence matrix