Row Equivalent Matrix for Homogeneous System has same Solutions

Theorem
Let:


 * $\mathbf A \mathbf x = \mathbf b$

be matrices representing a set of simultaneous linear equations, as presented in Solution to Simultaneous Linear Equations.

Suppose $\mathbf H$ is row equivalent to $\mathbf A$.

Then the solutions to $\mathbf H \mathbf x = \mathbf b$ are the same as the solutions to $\mathbf A \mathbf x = \mathbf b$:


 * $\mathbf A \sim \mathbf H \implies \left\{{\mathbf x: \mathbf A \mathbf x = \mathbf b}\right\} = \left\{{\mathbf x: \mathbf H \mathbf x = \mathbf b}\right\}$

where $\sim$ represents row equivalence.

Corollary

 * $\left\{{\mathbf x: \mathbf A \mathbf x = \mathbf b}\right\} = \left\{{\mathbf x: \operatorname{rref}\left({\mathbf A}\right) \mathbf x = \mathbf b}\right\}$

where $\operatorname{rref}\left({\mathbf A}\right)$ is the reduced row echelon form of $\mathbf A$.

Proof
Let:

be the system of equations to be solved.

Suppose the elementary row operation of multiplying by a non-zero scalar $a$ is performed.

Then the effect on the system of equations would be:

which clearly has the same solutions as the original equation.

Suppose the elementary row operation of adding a scalar multiple of one row to another row is performed.

Then the effect on the system of equations would be:

As both sides of equation $i$ are equal to each other, this operation is simply performing the same act on both sides of equation $j$.

Suppose the elementary row operation of row interchange was performed.

Then the systems of equations:

and:

clearly have the same solution set.

Hence the result, by the definition of row equivalence.

Proof of Corollary
Follows from the main result and from Matrix Row Equivalent to Reduced Echelon Matrix.