Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices

== Theorem ==
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$.
Let $\mathbf A \in \map \MM {m, n}$ be a matrix. 
Let $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Then there exists a unique nonsingular square matrix $\mathbf K$ of order $n$ such that:
:$\mathbf A \mathbf K = \mathbf B$
where $\mathbf K$ is the product of a finite sequence of elementary column matrices.
== Proof ==
The proof proceeds by induction.
By definition, $\Gamma$ is a finite sequence of elementary column operations on $\mathbf A$.
Let $\sequence e_k$ denote a finite sequence of elementary column operations $\tuple {e_1, e_2, \ldots, e_k}$ applied on $\mathbf A$ in order: first $e_1$, then $e_2$, then $\ldots$, then $e_k$.
Let $\Gamma_k$ be the column operation which consists of $\sequence e_k$.
Let $\mathbf E_k$ denote the elementary column matrix of order $n$ formed by applying $e_k$ to the unit matrix $I_n$.
For all $r \in \Z_{>0}$, let $\map P r$ be the proposition:
:For all $\Gamma_r$, there exists a unique nonsingular square matrix $\mathbf K_r$ of order $n$ such that:
::$\mathbf A \mathbf K_r = \mathbf B_r$
:where:
::$\Gamma_r$ is a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B_r \in \map \MM {m, n}$.
::$\mathbf K_r$ is the product of the finite sequence of elementary column matrices:
:::$\mathbf K_r = \mathbf E_1 \mathbf E_2 \dotsb \mathbf E_{r - 1} \mathbf E_r$
=== Basis for the Induction ===
$\map P 1$ is the case where $\Gamma_1$ is a single-term sequence consisting of one elementary column operation $e_1$.
Let $e_1$ be an elementary column operation operating on $\mathbf A$, which transforms $\mathbf A$ into $\mathbf B_1$.
By definition, there exists exactly one elementary column matrix $\mathbf E_1$ of order $m$ such that $\mathbf E_1$ is the result of applying $e_1$ to the unit matrix $\mathbf I$ of order $n$.
From the corollary to Elementary Column Operations as Matrix Multiplications:
:$\mathbf A \mathbf E_1 = \mathbf B_1$
By Elementary Column Matrix is Nonsingular, $E_1$ is nonsingular.
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:For all $\Gamma_k$, there exists a unique nonsingular square matrix $\mathbf K_k$ of order $n$ such that:
::$\mathbf A \mathbf K_k = \mathbf B_k$
from which it is to be shown that:
:For all $\Gamma_{k + 1}$, there exists a unique nonsingular square matrix $\mathbf K_{k + 1}$ of order $n$ such that:
::$\mathbf A \mathbf K_{k + 1} = \mathbf B_{k + 1}$
=== Induction Step ===
This is the induction step:
By definition, $\Gamma_{k + 1}$ is a column operation consisting of a finite sequence of elementary column operations $\tuple {e_1, e_2, \ldots, e_k, e_{k + 1} }$ applied on $\mathbf A$ in order.
Thus $\Gamma_{k + 1}$ consists of the finite sequence of elementary column operations $\tuple {e_1, e_2, \ldots, e_k}$ applied on $\mathbf A$ in order, followed by a further elementary column operation $e_{k + 1}$.
By the induction hypothesis, there exists a unique nonsingular square matrix $\mathbf K_k$ of order $m$ such that:
:$\mathbf A \mathbf K_k = \mathbf B_k$
where $\mathbf B_k \in \map \MM {m, n}$ is the result of applying $\sequence e_k$ to $\mathbf A$ in order.
Let $e_{k + 1}$ be applied to $\mathbf B_k$.
By definition, there exists exactly one elementary column matrix $\mathbf E_{k + 1}$ of order $m$ such that $\mathbf E_{k + 1}$ is the result of applying $e_{k + 1}$ to the unit matrix $\mathbf I$ of order $m$.
Then:
By Product of Matrices is Nonsingular iff Matrices are Nonsingular, $\mathbf K_k \mathbf E_{k + 1}$ is nonsingular.
We have that $\mathbf K_k$ is the unique nonsingular square matrix resulting from the application of $\sequence e_k$ on $\mathbf I_m$.
Thus $\mathbf K_k \mathbf E_{k + 1}$ is the unique nonsingular square matrix resulting from the application of $\sequence e_{k + 1}$ on $\mathbf I_m$.
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore, for every column operation $\Gamma$ which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$, there exists a unique nonsingular square matrix $\mathbf R$ of order $m$ such that:
:$\mathbf A \mathbf K = \mathbf B$
where:
:$\mathbf K$ is the product of a finite sequence of elementary column matrices.
== Also see ==
* Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations
Category:Column Operations
Category:Proofs by Induction