TITLE: Understanding matrices as linear transformations & Relationship with Gaussian Elimination and Bézout's Identity
QUESTION [7 upvotes]: I am currently taking a intro course to abstract algebra and am revisiting ideas from linear algebra so that I can better understand examples. 
When i was in undergraduate learning L.A., I thought of matrix manipulations as ways of solving $n \times n$ systems of equations. Recently i was exposed to the idea of a matrix being a linear transformation, and matrix multiplication being composition of linear transformations. Im trying to understand this in a more intuitive way and was hoping for some insight...
I was thinking of a basic $2\times2$ example and how it affects a point $(x,y)$. 
We could have a matrix :
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
When we 'apply' or multiply this to a point $(x,y)$ using matrix multiplication we get new $x' = ax + by$ and $y' = cx + dy$.
So if $b,c = 0$, then I can see that what we are doing is 'scaling' both $x \;\& \;y$. I'm guessing that if $b,c \neq 0$, then this becomes some sort of rotation or reflection, but how do you understand this on a fundamental level?
How do these operations relate to Gaussian elimination when we are trying to solve systems of equations? Or are are these two seperate applications of matrices?
Another observation is that when multiplying a matrix such as this one with a point, we get two equations which remind me of Bézout's identity. Am I overanalyzing this or can I draw connections between these two concepts?
Thanks for any input!

REPLY [2 votes]: First, matrices can represent linear transformations as in $ x \mapsto Ax$, but they can also represent bilinear forms: $<x,y> = x^T A y$.  But here we will stick to the linear transformation case. What you should do is search for some insight by looking fos some canonical examples. 
Rotation matrices: these are implementing rotations. For the simplest case, the plane, we have $R=\left(\begin{smallmatrix} \cos(\theta) & -\sin(\theta) \\ 
      \sin(\theta) & \cos(\theta) \end{smallmatrix}\right)$. You should try some examples and plot them! Then we have reflection matrices:  You can find a derivation of a 2D reflection matrix here: http://www.scibuff.com/2009/06/22/reflection-matrix/
Together, rotation and reflection matrices are called orthogonal matrices. Rotations have determinant $1$, and reflections have determinant $-1$. They satisfy $Q^T Q= Q Q^T = I$ where $I$ is the identity matrix. To see they have determinant $\pm 1$, calculate $1=\det(I)=\det(Q^T Q) =\det(Q)^2$ and solve!
A very simple example is in $\mathbb R^3$, a rotation with rotation axes equal to the vertical ($z$) axes, and rotating an angle $\theta$ in the $xy$-plane:
$$
\begin{pmatrix}  \cos(\theta) & -\sin(\theta) & 0 \\
                 \sin(\theta) & \cos(\theta) & 0 \\
                    0   &   0  & 1  \end{pmatrix}
$$
Again, you should calculate an draw some examples!
In mechanics and engineering diciplines the shear matrices are important: Let
$$ S = \begin{pmatrix}  1 & 0.5  \\
                        0 & 1 \end{pmatrix}
$$
This effects a shear: You should compute some examples, and you will see what it means. More examples to come ...