Shearing \(\mathbb{R}^2\) Parallel to the x-Axis

The shear transformation \(S\) of \(\RNrSpc{2}\), parallel to the \(x\)-axis with shear vector \(\Vect{s} = (1,0)\) is given by.

\[ S\from \RNrSpc{2}\longrightarrow \RNrSpc{2},\qquad S(x,y) = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{matrix}{c} x \\ y \end{matrix} \right] \]

The transformation effect of \(S\) on the square spanned by the vectors \(\StdBss{1}\) and \(\StdBss{2}\) is illustrated in the image below.

More generally, if \(\Vect{s} = (a,0)\) is an arbitrary shear vector for a shear transformation \(S\) of \(\RNrSpc{2}\) parallel to the \(x\)-axis, then \(S\) is given by

\[ S(x,y) = \left[ \begin{array}{rr} 1 & a \\ 0 & 1 \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] \]

In particular, \(S\) leaves \(\StdBssVec{1}\) unchanged, and it shears \(\StdBssVec{2}\) into \((0,1) + (a,0) = (a,1)\).