Row Equivalent Matrix for Homogeneous System has same Solutions

Theorem
Let $\mathbf A$ be a matrix in the matrix space $\mathbf M_{m,n}\left({\R}\right)$ such that:


 * $\mathbf A \mathbf x = \mathbf 0$

represents a homogeneous system of linear equations.

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

Then the solution set of $\mathbf H \mathbf x = \mathbf 0$ equals the solution set of $\mathbf A \mathbf x = \mathbf 0$.

That is:


 * $\mathbf A \sim \mathbf H \implies \left\{{\mathbf x: \mathbf A \mathbf x = \mathbf 0}\right\} = \left\{{\mathbf x: \mathbf H \mathbf x = \mathbf 0}\right\}$

where $\sim$ represents row equivalence.

Corollary

 * $\left\{{\mathbf x: \mathbf A \mathbf x = \mathbf 0}\right\} = \left\{{\mathbf x: \operatorname{rref}\left({\mathbf A}\right) \mathbf x = \mathbf 0}\right\}$

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 one row $i$ by a non-zero scalar $\lambda$ is performed.

Recall, the $i$th row of the matrix represents the $i$th equation of the system to be solved.

Then, this is logically equivalent to multiplying the $i$th equation on both sides by the scalar $lambda$:

which clearly has the same solutions as the original equation.

Suppose the elementary row operation of adding a scalar multiple of row $i$ to another row $j$ is performed.

Recall, the $i$th and $j$th row of the matrix represent the $i$th and $j$th equation in the system to be solved.

Thus this is logically equivalent to manipulating the $i$th and $j$th equations as such:

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

This clearly will have no effect on the solution set of the system of equations.

Suppose the elementary row operation of interchanging row $i$ and row $j$ is performed.

Recall, the $i$th and $j$th row of the matrix represent the $i$th and $j$th equation in the system to be solved.

Then, interchanging row $i$ and row $j$ is logically equivalent to switching the $i$th equation and the $j$th equation of the system to be solved.

But clearly the system containing the following two equations:

has the same solution set as a system instead containing the following two equations:

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.