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kSquared

Matrix : linear :: (?) : nonlinear

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Consider the following statement: "For every linear transformation, there's exactly one corresponding matrix, and for every matrix there is a unique linear transformation." Is there an equivalent for nonlinear transformations? That is, can this blank be filled in? "For every nonlinear transformation, there's exactly one corresponding [ ? ], and for every [ ? ] there is a unique nonlinear transformation."

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Linear transformations are completely determined by the image of the vectors of a base (basis?). That's why you can describe them with as little information as a single matrix.

If by non-linear you mean an arbitrary mapping from R^n to R^m, there are too many such functions to characterize them with any object that is any simpler.

If you are interested in algebraic or analytic transformations, then there is some hope (a few polynomials or a few power series will do the job).

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Exactly as alvaro said.

For example, image defines usually nonlinear(piecewise linear, for example) 2D->3D mapping, where resulting coordinate is color of pixel...

Some specific nonlinear transformations, for example perspective transformation can be somehow storen in matrix. Perspective transform is given by ratio of 2 linear transforms, so matrix is used to linearly transform some vector that is then divided by w component, so we get ratio of 2 linear transforms...

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