| # Linear Maps Matrices > Basics | Matrix products | Special matrices | QR | Matrix inverses | Linear maps | Matrix norms | Applications Definitions and interpretation First-order approximation of non-linear maps ## Definition and Intepretation ### Definition A map is linear (resp. affine) if and only if every one of its components is. The formal definition we saw here for functions applies verbatim to maps. To an matrix , we can associate a linear map , with values . Conversely, to any linear map, we can uniquely associate a matrix which satisfies for every . Indeed, if the components of , , , are linear, then they can be expressed as for some . The matrix is the matrix that has as its -th row: Hence, there is a one-to-one correspondence between matrices and linear maps. This is extending what we saw for vectors, which are in one-to-one correspondence with linear functions. This is summarized as follows. Representation of affine maps via the matrix-vector product. A function is affine if and only if it can be expressed via a matrix-vector product: for some unique pair , with and . The function is linear if and only if . The result above shows that a matrix can be seen as a (linear) map from the ‘‘input“ space to the ‘‘output” space . Both points of view (matrices as simple collections of vectors, or as linear maps) are useful. ### Interpretations Consider an affine map . An element gives the coefficient of influence of over . In this sense, if we can say that has much more influence on than . Or, says that does not depend at all on . Often the constant term is referred to as the ‘‘bias’’ vector. ## First-order approximation of non-linear maps Since maps are just collections of functions, we can approximate a map with a linear (or affine) map, just as we did with functions here. If is differentiable, then we can approximate the (vector) values of near a given point by an affine map : where is the derivative of the -th component of with respect to . ( is referred to as the Jacobian matrix of at.) Examples: Navigation by range measurement. State-space models of dynamical systems.