Elementary Row Operations

Definition: Elementary Row Operations

Elementary row operations are the following manipulations which can be done on a system of linear equations without changing its solutions:

• Exchanging two rows (row swap)
• Multiplying a row by a constant $\lambda \ne 0$
• Adding a non-zero multiple of one row to another row

Theorem: Elementary Matrix

Each elementary row operation can by represented as the matrix product of the augmented matrix $(A|\vec{b})\in K^{m\times n}$ of a system of linear equations with a matrix $E$, which is obtained by applying the row operation to the identity matrix $I_m$.

$$E\cdot (A|\vec{b})$$

PROOF

TODO

EXAMPLE

TODO