General form of a shear map From Wikipedia > In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction. I'm interested in the matrix representation of a general shear map in the plane. Every resource I look at either only gives the horizontal and vertical shear matrices $$\begin{bmatrix} 1 & k \\\ 0 & 1\end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 1 & 0 \\\ k & 1\end{bmatrix}$$ or a couple have said that $$\begin{bmatrix} 1 & a \\\ b & 1\end{bmatrix}$$ is also a shear map. However I don't think that last one is if neither $a$ nor $b$ is zero because, as far as I understand, shear mappings should be area preserving. So then what is the general form of the matrix representing a shear map that displaces all vectors in the direction parallel to an arbitrary vector $(x,y)$?

Lets say generalized sheer takes the vector $\begin{bmatrix} x\\\y\end{bmatrix}$ to $\begin{bmatrix} x\\\y\end{bmatrix}$ and the orthogonal vector,

$\begin{bmatrix} -y\\\x\end{bmatrix}$ to $\begin{bmatrix} -y + kx\\\x + ky\end{bmatrix}$

in term of the basis$\left\\{\begin{bmatrix} x\\\y\end{bmatrix},\begin{bmatrix} -y\\\x\end{bmatrix}\right\\}$ our transformation is $T = \begin{bmatrix} 1&k\\\0&1 \end{bmatrix}$

And in terms of the standard basis

$\begin{bmatrix} 1 -\frac {kxy}{x^2+y^2} & \frac {kx^2}{x^2+y^2}\\\\-\frac {ky^2}{x^2+y^2}&1+\frac{kxy}{x^2+y^2}\end{bmatrix}$