Proof of geometric congruence using linear algebra We may assume some set $T$ of all triangles within the same plane. Let $R$ be defined on $T$ where $a\ R\ b$ if the triangles $a, b$ are congruent. We may assume  congruence to be defined as follows 

a triangle may be transformed into the other by some combination of reflections,  rotations, and translations. 

It appears we must first define what reflection means. I am trying to find a very straightforward definition of what "reflection" means for triangles; it seems to imply  some kind of transformation $f$ on an element $a$ such that $(f \circ f)(a) =f(f(a)) = a$ and $f(a) \ne a$. But first, it seems there has to be a mathematically precise way to show the following hypothesis: 

any triangle $A$ can be represented two linearly independent vectors
  $u, v \in \mathbb{R}^2$ such that $u, v$ represent two of its sides.  

That is, I want to define "reflection" with some basic linear algebra definitions instead of assuming geometric facts like SAS. It seems that I will have to somehow show that any triangle $A$ can be sufficiently represented by a linearly independent set $\left[u\ \ \ v \right ] $ on $\mathbb{R}^2$. 
If so, then there exists some transformation $f: \mathbb{R}^2 \to \mathbb{R}^2$ such that $(f \circ f)(A) = f(f(A)) = A$ for $A \ne f(A)$, namely, $ (f \circ f)(A) = A \cdot \mathbf{I} = A  \cdot (A^{-1}A)= A \cdot (A A^{-1})$. Therefore, we have a precise way of explaining what reflection is, and that any reflection of a triangle $A$ may be transformed back into $A$ with a sequence of reflections.  
This may not be the simplest approach to the problem of defining reflection, and in turn, to show precisely what this definition of congruence means. I still would like to persist with this basic linear algebra approach and ask, what is missing to show reflection here?
 A: A reflection about any plane $H$ can be realized from a reflection about the $e_1\times \dots \times e_k$-plane (in 2-D this would be $e_1$-axis)  after some rotation and/or translation.  Therefore simply define reflection about the plane by taking $-e_n$.  In 2-D this would be $f(x,y) = (x, -y)$.
You can actually define reflection about any plane, but done with the basis you're working in seems the most convenient.
A: Any congruence of the plane is a composition of at most $3$ reflections, which is related to the fact that the composition of $2$ reflections is a rotation or a translation.  You do not actually need to define rotations and translations separately under your definition of congruence.
Linear algebra is not enough to define congruence without additional nonlinear structure such as an orthogonality relation, a distance measure, or a dot product of vectors.  The linear structure alone gives you only affine transformations $v \to Av + b$. The closest thing to reflections that can be defined from vector space structure is (affine) involutions, linear transformations such that $T \circ T$ is the identity.  The involution is called nontrivial if $T$ is not the identity transformation.
An affine involution, geometrically, can be defined by taking any two intersecting lines as axes of a coordinate system and sending $(x,y)$ to $(x,-y)$.  
A reflection is an involution that preserves any of the nonlinear additional structures that were listed previously.
