affine basis in a sentence

1. Equivalently, } is an affine basis of an affine space if and only } is a linear basis of the associated vector space.
2. A set of at most d + 1 points in general linear position is also said to be " affinely independent " ( this is the affine analog of linear independence of vectors, or more precisely of maximal rank ), and d + 1 points in general linear position in affine " d "-space are an affine basis.
3. This use of the standard ( n-1 )-simplex and " f "-orthant as standard objects that map to a polytope or that a polytope maps into should be contrasted with the use of the standard vector space K ^ n as the standard object for vector spaces, and the standard affine hyperplane \ { ( x _ 0, \ ldots, x _ n ) \ mid \ sum x _ i = 1 \ } \ subset K ^ { n + 1 } as the standard object for affine spaces, where in each case choosing a linear basis or affine basis provides an " isomorphism, " allowing all vector spaces and affine spaces to be thought of in terms of these standard spaces, rather than an onto or one-to-one map ( not every polytope is a simplex ).
4. It's difficult to find affine basis in a sentence.