Row equivalence of matrices

Dependencies:

1. Elementary row operation
2. Every elementary row operation has a unique inverse
3. /sets-and-relations/equivalence-relation
4. Field

Let $A$ and $B$ be matrices of the same size. $A$ and $B$ are row equivalent iff $B$ can be obtained from $A$ by applying a finite number of elementary row operations to $A$.

Row equivalence is an equivalence relation when matrices are over a field.

Proof

• Reflexive: $A$ is row equivalent to $A$, since $A$ can be obtained from $A$ by applying 0 elementary row operations.

• Symmetric: If $B$ can be obtained from $A$ by applying a finite number of row operations $[R_1, R_2, \ldots, R_n]$, then $A$ can be obtained from $B$ by applying the operations $[R_n^{-1}, R_{n-1}^{-1}, \ldots, R_1^{-1}]$. Here $R_i^{-1}$ is the inverse operation of $R_i$.

• Transitive: If $B$ can be obtained from $A$ by applying the operations $[R_1, R_2, \ldots, R_m]$ and $C$ can be obtained from $B$ by applying the operations $[S_1, S_2, \ldots, S_n]$, then $C$ can be obtained from $A$ by applying the operations $[R_1, R_2, \ldots, R_m, S_1, \ldots, S_n]$.

Therefore, row equivalence is an equivalence relation.

Dependency for:

1. Row equivalence matrix
2. Row equivalent matrices have the same row space
3. Equations with row equivalent matrices have the same solution set

Info:

• Depth: 5
• Number of transitive dependencies: 8

Transitive dependencies:

1. /sets-and-relations/equivalence-relation
2. Group
3. Ring
4. Field
5. Semiring
6. Matrix
7. Elementary row operation
8. Every elementary row operation has a unique inverse