Vector-valued function
Linear case

In the linear case the function can be expressed in terms of matrices:

where y is an n × 1 output vector, x is a k × 1 vector of inputs, and A is an n × k matrix of parameters. Closely related is the affine case (linear up to a translation) where the function takes the form

where in addition b'' is an n × 1 vector of parameters.

The linear case arises often, for example in multiple regression, where for instance the n × 1 vector y ^ {\displaystyle {\hat {y}}} of predicted values of a dependent variable is expressed linearly in terms of a k × 1 vector β ^ {\displaystyle {\hat {\boldsymbol {\beta }}}} (k < n) of estimated values of model parameters:

in which X (playing the role of A in the previous generic form) is an n × k matrix of fixed (empirically based) numbers.
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