visual intuition for linear transformation I know that when a transformation is linear, it satisfies the following properties:     $$T(cu+dv) = cT(u) + dT(v)$$
But, Given a arbitrary transformation T, and some vectors u and v. How can we visualize that the transformation is linear, intuitively? What is the visualization of a transformation that is linear, based on the mathematical property.
 A: A linear transformation preserves addition and scalar multiplication. That means if you stretch a vector the image will stretch by the same factor. Preserving the addition means the resultant of two vectors will map to the resultant of the images. Think of projection of a 3 dimensional object on a plane , or taking pictures of an object.The picture of a triangle is a triangle and if you take a picture of two triangles where one is twice the other in size, the picture shows the same proportion.          
A: Let's try to translate the mathematical statement $T(cu + dv) = cT(u) + dT(v)$ into a visualization. Given your vectors $u$ and $v$, imagine a parallelogram grid whose points are $0$, $u$, $v$, $u+v$, $2u$, $2v$, $2u+v$, $2v+u$, $2u+2v$, etc. Upon transformation by $T$, we should get $0$, $T(u)$, $T(v)$, $T(u)+T(v)$, $2T(u)$, etc., i.e., another parallelogram grid (possibly degenerate).
That's one visual characterization of a linear transformation: it transforms a parallelogram grid into a parallelogram grid.
