Affine Space - Examples

Examples

  • 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.
  • Any coset of a subspace of a vector space is an affine space over that subspace.
  • If is a matrix and lies in its column space, the set of solutions of the equation is an affine space over the subspace of solutions of .
  • The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation.
  • Generalizing all of the above, if is a linear mapping and y lies in its image, the set of solutions to the equation is a coset of the kernel of, and is therefore an affine space over .

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