Hence affine combinations are defined in vector spaces over any field.

At these points, the defining formula has zero value, and therefore will also equal zero for any affine combination.

Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

An affine combination is a linear combination in which the sum of the coefficients is 1.

Affine combinations are like convex combinations, but the coefficients are not required to be non-negative.

Thus the predicted class is an affine combination of the classes of every other point, weighted by the softmax function for each where is now the entire transformed data set.

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations.

Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively.

In AA, the quantities of interest are represented as affine combinations (affine forms) of certain primitive variables, which stand for sources of uncertainty in the data or approximations made during the computation.

When a stochastic matrix, A, acts on a column vector, B, the result is a column vector whose entries are affine combinations of B with coefficients from the rows in A.

By restricting the coefficients used in linear combinations, one can define the related concepts of affine combination, conical combination, and convex combination, and the associated notions of sets closed under these operations.

While Alice knows the "linear structure", both Alice and Bob know the "affine structure"-i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1.

Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others.

If however one puts no restrictions at all on the numbers , instead of an affine combination one has a linear combination, and the resulting set is the linear span of S, which contains the affine hull of S.

Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over an ordered field (or ordered ring), generally the real numbers.

The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace.

If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all be non-negative, one obtains the convex hull of S, which cannot be larger than the affine hull of S as more restrictions are involved.

In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of , so the set of fixed points of forms an affine subspace (in 3D: a line or a plane, and the trivial cases, a point or the whole space).

If, on the other hand, the kernel assumes negative values, such as the sinc function, then the value of the filtered signal will instead be an affine combination of the input values, and may fall outside of the minimum and maximum of the input signal, resulting in undershoot and overshoot.