Elementary Row Operation

Last updated Nov 1, 2022

Let $F$ be a Field and let $m \in \mathbb{N}$. Let $F^{m \times \cdot} = \bigcup\limits_{n \in \mathbb{N}} F^{m \times n}$. An Elementary Row Operation is one of the following 3 types of Functions on $F^{m \times \cdot}$:

1. Let $c \in F \setminus {0}$ and $r \in [m]$. Then $e: F^{m \times \cdot} \to F^{m \times \cdot}$ defined as $$e(A){ij} = \begin{cases} A{ij} & \text{if } i \neq r \\ cA_{rj} & \text{if } i = r \\ \end{cases}$$ is an Elementary Row Operation (known as a row scaling)
2. Let $c \in F$ and $r, s \in [m]$. Then $e: F^{m \times \cdot} \to F^{m \times \cdot}$ defined as $$e(A){ij} = \begin{cases} A{ij} & \text{if } i \neq r \\ A_{rj} + cA_{sj} & \text{if } i = r \\ \end{cases}$$ is an Elementary Row Operation (known as a row addition).
3. Let $c \in F$ and $r, s \in [m]$. Then $e: F^{m \times \cdot} \to F^{m \times \cdot}$ defined as $$e(A){ij} = \begin{cases} A{ij} & \text{if } i \neq r, i \neq s \\ A_{sj} & \text{if } i = r \\ A_{rj} & \text{if } i = s \end{cases}$$ is an Elementary Row Operation (known as a row swap).

1. Matrix Representation of Elementary Row Operations
2. We represent the Set of Elementary Row Operations on $F^{m \times \cdot}$ as $\text{Row}F^{m \times \cdot}$.
3. TODO this has some relationship to Groups. However, I can’t really talk about Matrix Inverses without setting up details on Vector Space Basis. I need some details about Matrix Equation Systems to establish that.