Linear Functions and Maps
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
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 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 InterpretationsThe interpretation of
Example: Beer-Lambert law in absorption spectrometry. Linear and affine mapsDefinitionA map To an Indeed, if the components of This is summarized as follows. Theorem: Representation of affine maps via the matrix-vector product.
A function The theorem shows that a matrix can be seen as a (linear) map from the ‘‘input“ space InterpretationsConsider an affine map First-order approximation of non-linear functionsMany maps 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 Multi-dimensional caseWith more than one variable, we have a similar result. Let us approximate a differentiable function The approximate function Theorem: First-order expansion of a function.
The first-order approximation of a differentiable function The case of mapsThe above can be extended to maps. If Example: Navigation by range measurement. Other sources of linear modelsChange of variablesLinearity can arise from a simple change of variables. This is best illustrated with a specific example. Example: Power laws. Linear dynamical systemsMany linear systems arise from models of dynamical systems, which describe how the state of a system evolves over time. The state is in general a vector fully describing the system at a given time (comprising say the temperature at several locations of a furnace, or the positions and velocities of several locations on a bridge, or various economic indicators, etc). A linear dynamical model postulates that the state at the next instant is a linear function of the state at past instants, and possibly other ‘‘exogeneous’’ inputs. Such systems can result from the linearization of a non-linear dynamical model. Example: Population dynamics. |