Answer to Question #163167 in Linear Algebra for Geek

Question

1.What is affine space, affine set and affine varieties.Explain with examples.

2.Define a relation between affine space and vector space with examples

Answer 1

An affine space is an abstraction of how geometrical points (in the plane, say) behave.

When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space

A set A is said to be an affine set if for any two distinct points, the line passing through these points lie in the set A.

Affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space kn of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.

2.An affine space A is a set of elements with a difference function. This difference is a binary function, which takes two points p and q (both in A) and yields an element (a vector) v of a vector space V (for each unique A, there is an unique V, which is the vector space associated to A). We write v=p−q.