Linear FunctionsVectors > Basics | Scalar product, Norms | Projection on a line | Orthogonalization | Hyperplanes | Linear functions | Application
Linear and affine functionsDefinitionLinear functions are functions which preserve scaling and addition of the input argument. Affine functions are ‘‘linear plus constant’’ functions. Formal definition, linear and affine functions. A function
A function An alternative characterization of linear functions is given here. Examples: Consider the functions
The function Connection with vectors via the scalar productThe following shows the connection between linear functions and scalar products. Theorem: Representation of affine function via the scalar product:
A function ![]() for some unique pair The theorem shows that a vector can be seen as a (linear) function from the ‘‘input“ space Gradient of an affine functionThe gradient of a function An affine function ![]() Example: gradient of a linear function. InterpretationsThe interpretation of
Example: Beer-Lambert law in absorption spectrometry. First-order approximation of non-linear functionsMany functions are non-linear. A common engineering practice is to approximate a given non-linear map with a linear (or affine) one, by taking derivatives. This is the main reason for linearity to be such an ubiquituous tool in Engineering. One-dimensional caseConsider a function of one variable ![]() where Multi-dimensional caseWith more than one variable, we have a similar result. Let us approximate a differentiable function The approximate function ![]() where ![]() where Theorem: First-order expansion of a function.
The first-order approximation of a differentiable function ![]() where Example: a linear approximation to a non-linear function. Other sources of linear modelsLinearity can arise from a simple change of variables. This is best illustrated with a specific example. Example: Power laws. |