Matrix transform vs linear transform

[blixtra]

Every matrix transformation from R-n to R-m is a linear transformation. The converse of this is not true: every linear transformation is not a matrix transformation.

According to David Lay's text, Linear Algebra, the term matrix transformation describes how a mapping is implemented, while the term linear transformation focus on a property of the mapping. His text is replete with examples of matrix transforms which are linear tranforms, but silent on examples of linear transforms that are not matrix transforms.

Where can I find a counterexample that illustrates a linear transformation which is not a matrix tranformation?

Also, as I contemplate future posts, is there a link that explains how to format my questions, equations, etc in LyX or something similar? Thanks.

 

[Defennder]

If Wikipedia is to be believed, here is something you can use:

Linear[/PLAIN] transformations are not the only ones that can be represented by matrices. Using homogeneous coordinates, both affine transformations and perspective projections on Rn can be represented as linear transformations on RPn+1 (that is, n+1-dimensional real projective space). For this reason, 4x4 transformation matrices are widely used in 3D computer graphics.

3-by-3 or 4-by-4 transformation matrices containing homogeneous coordinates are often called, somewhat improperly, "homogeneous transformation matrices". However, the transformations they represent are, in most cases, definitely non-homogeneous and non-linear (like translation, roto-translation or perspective projection). And even the matrices themselves look rather heterogeneous, i.e. composed of different kinds of elements (see below). Since they are multi-purpose transformation matrices, capable of representing both affine and projective transformations, they might be called "general transformation matrices", or, depending on the application, "affine transformation" or "perspective projection" matrices. Moreover, since the homogeneous coordinates describe a projective vector space, they can also be called "projective space transformation matrices".

Hope that helps.