In this shear transformation of an image of the Mona Lisa, the picture was deformed in such a way that its central vertical axis was not modified. (Note: The corners have been cropped on the right hand picture.)

In mathematics, a shear is a particular kind of linear mapping. For a vector space V and subspace W, a shear fixing W translates all vectors parallel to W. Image File history File links Eigen. ... Image File history File links Eigen. ... Mona Lisa, or La Gioconda, is a 16th Century oil painting on poplar wood by Leonardo da Vinci and is one of the most famous paintings in Western art history; few other works of art are as romanticized, celebrated, or reproduced. ... Euclid, detail from The School of Athens by Raphael. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

To be more precise, if V is the direct sum of W and W′, and we write vectors as In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

v = w + w′

correspondingly, the typical shear fixing W is L where

L(v) = w + (w′ + M(w′))

where M is a linear mapping from W′ into W. Therefore in block matrix terms L can be represented as 2×2 , with blocks on the diagonal I (identity matrix), with M above the diagonal, and 0 below. In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

Mathematical shears are also called transvections.