Row equivalent matrices have the same row space

Dependencies:

  1. Field
  2. Row equivalence of matrices
  3. Row space
  4. Every elementary row operation has a unique inverse

Let $A$ and $B$ be 2 matrices over a field. Then $A$ and $B$ are row-equivalent implies that $A$ and $B$ have the same row space.

Proof

Suppose $B$ is obtained from $A$ via an elementary row operation $R$. Then each row of $B$ is a linear combination of the rows of $A$.

Let $x$ be an element in the row space of $B$. $x$ is a linear combination of the rows of $B$, which are linear combinations of the rows of $A$. Therefore, $x$ is a linear combination of the rows of $A$. Therefore, $x$ is in the row space of $A$. Therefore, row space of $B$ is a subset of the row space of $A$.

Since, every elementary row operation in a field has an inverse, $A$ can be obtained from $B$ via the elementary row operation $R^{-1}$. Therefore, row space of $A$ is a subset of the row space of $B$.

This means that if one matrix can be obtained by an elementary row operation on the other matrix, then those matrices have the same row space. Therefore, any 2 matrices which are row equivalent have the same row space.

Dependency for:

  1. RREF is unique
  2. A matrix is full-rank iff its rows are linearly independent
  3. Rank of a matrix

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /sets-and-relations/equivalence-relation
  3. Group
  4. Ring
  5. Field
  6. Vector Space
  7. Semiring
  8. Matrix
  9. Matrices over a field form a vector space
  10. Row space
  11. Elementary row operation
  12. Every elementary row operation has a unique inverse
  13. Row equivalence of matrices