It seems to me that any linear transformation in $R^{n\times m}$ is just a series of applications of rotation(actually i think any rotation can be achieved by applying two reflections, but not sure), reflection, shear, scaling and projection transformations. One or more of each kind in some order.
This is how I have been imagining it to myself, but I was unable to find proof of this on the internet.
Is this true? And if this is true, is there a way to find such a decomposition?
EDIT: to make it clear, I am asking whether it is true that $\forall A\in R^{n \times m} $,$A=\prod_{i=1}^{k}P_i$ Where $P_i$ is rotation, reflection, shear, scaling, or projection matrix in $R^{n_i\times m_i}$. Also $n,m,k\in N$,and $n_i,m_i\in N$ for all i.
And if it is true then how can we decompose it into that product.