In mathematics, a shear mapping or transvection is a particular kind of linear mapping. Its effect leaves fixed all points on one axis and other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from the axis. It is notable that shear mappings carry areas into equal areas.
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In the plane {(x, y): x,y ∈ R }, a horizontal shear (or shear parallel to the x axis) is represented by the linear mapping
This leaves horizontal lines y = c invariant, but for m ≠ 0 maps vertical lines x = a into lines y' = (x' − a)/m having slope 1/m
Substituting 1/m for m in the matrix gives lines y = m(x − a) of slope m, if desired.
A vertical shear (or shear parallel to the y axis) of lines is accomplished by the linear mapping
The vertical shear leaves vertical lines x = a invariant, but maps horizontal lines y = b into lines y' = mx' + b
The matrices above are special cases of shear matrices, which allow for generalization to higher dimensions. The shear elements here are either m or 1/m, case depending.
The area-preserving property of a shear mapping can be used for results involving area. For instance, the Pythagorean theorem has been illustrated with shear mapping (see external link).
For a vector space V and subspace W, a shear fixing W translates all vectors parallel to W.
To be more precise, if V is the direct sum of W and W ′, and we write vectors as
correspondingly, the typical shear fixing W is L where
where M is a linear mapping from W ′ into W. Therefore in block matrix terms L can be represented as
with blocks on the diagonal I (identity matrix), with M above the diagonal, and 0 below.
  | Pythagorean theorem by shear mapping - Pythagoras Proof - GeoGebra Dynamic Worksheet |
| | "Shear" - Shear -- from Wolfram MathWorld |
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