Let V be a vector space over a field K, and let A be a nonempty set. Now define addition p + a element A for any vector a element V and element p element A subject to the conditions: 1.p + 0 = p. 2.(p + a) + b = p + (a + b). 3. For any q element A, there exists a unique vector a element V such that q = p + a. Here, a, b element V. Note that (1) is implied by (2) and (3). Then A is an affine space and K is called the coefficient field.

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AffineSpace