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- The affine span of vectors
*a*_{1},*a*_{2}, ... ,*a*_{r}in**R**^{n}is the set

- {
*λ*_{1}*a*_{1}+ ... +*λ*_{r}*a*_{r}|*λ*_{i}∈**R**,*λ*_{1}+ ... +*λ*_{r}= 1}.

- Prove that it is an affine subspace (i.e. a translated linear subspace).
- Points
*a*_{1},*a*_{2}, ... ,*a*_{r}in**R**^{n}are called*affinely independent*if whenever*λ*_{1}**a**_{1}+ ... +*λ*_{r}**a**_{r}= 0 with*λ*_{1}+ ... +*λ*_{r}= 0 then*λ*_{1}= ... =*λ*_{r}= 0.

Prove that for such points the set {*a*_{2}-*a*_{1},*a*_{3}-*a*_{1}, ... ,*a*_{r}-*a*_{1}} is linearly independent.

Prove that for such points the set of vectors { (*a*_{1}, 1), (*a*_{2}, 1), ... , (*a*_{r}, 1) } in**R**^{n}×**R**is linearly independent.

Prove that there is a*unique*affine map taking any (*n*+ 1) affinely independent points in**R**^{n}into any other (*n*+ 1) affinely independent points.- Find a group of (
*n*+ 1) × (*n*+ 1) matrices isomorphic to the affine group*A*(**R**^{n}).

[Look at Exercises 3 Question 3 to see the same result for*I*(**R**^{n}).]

Hence prove (again!) the result of Question 2, that there is a unique affine map taking any (*n*+ 1) affinely independent points in**R**^{n}into any other (*n*+ 1) affinely independent points.- If
*f*is an affine map, prove that*f*maps the affine span of**a**_{1},**a**_{2}, ... ,**a**_{r}to the affine span of*f*(*a*_{1}),*f*(*a*_{2}), ... ,*f*(*a*_{r}) and in fact, if*λ*_{1}+ ... +*λ*_{r}= 1 then*f*(*λ*_{1}*a*_{1}+ ... +*λ*_{r}*a*_{r}) =*λ*_{1}*f*(*a*1) + ... +*λ*_{r}*f*(*a*_{r}). Deduce that an affine map takes the centroid of any set to the centroid of its image.- If
*ABCD*is a quadrilateral in**R**^{2}, prove that the midpoints of its sides form a parallelogram whose diagonals meet at the centroid of its vertices.

Is this true for a quadrilateral in**R**^{3}also?- Prove that any similarity transformation which is not an isometry has a fixed point.
- If
*AB*and*A*'*B*' are two line segments in**R**^{2}, prove that there are two similarity transformations mapping*AB*to*A*'*B*' -- one of them direct (i.e. with linear part having positive determinant) and the other opposite.

[Hint: Consider squares with*AB*and*A*'*B*' as sides and consider affine maps taking one square to the other.] - {

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