Exercises 7

1. The affine span of vectors a₁, a₂, ... , a_r in Rⁿ is the set
   
   {λ₁a₁ + ... + λ_ra_r | λ_i ∈ R, λ₁ + ... + λ_r = 1}.
   
   Prove that it is an affine subspace (i.e. a translated linear subspace).

   Solution to question 1

   Points a₁, a₂, ... , a_r in Rⁿ are called affinely independent if whenever λ₁a₁ + ... + λ_ra_r = 0 with λ₁ + ... + λ_r = 0 then λ₁ = ... = λ_r = 0.
   Prove that for such points the set { a₂ - a₁, a₃ - a₁, ... , a_r - a₁ } is linearly independent.
   Prove that for such points the set of vectors { (a₁, 1), (a₂, 1), ... , (a_r , 1) } in Rⁿ × R is linearly independent.
   Prove that there is a unique affine map taking any (n + 1) affinely independent points in Rⁿ into any other (n + 1) affinely independent points.

   Solution to question 2

   Find a group of (n + 1) × (n + 1) matrices isomorphic to the affine group A(Rⁿ).
   [Look at Exercises 3 Question 3 to see the same result for I(Rⁿ).]
   Hence prove (again!) the result of Question 2, that there is a unique affine map taking any (n + 1) affinely independent points in Rⁿ into any other (n + 1) affinely independent points.

   Solution to question 3

   If f is an affine map, prove that f maps the affine span of a₁, a₂, ... , a_r to the affine span of f(a₁), f(a₂), ... , f(a_r) and in fact, if λ₁ + ... + λ_r = 1 then f(λ₁a₁ + ... + λ_ra_r) = λ₁f(a1) + ... + λ_rf(a_r). Deduce that an affine map takes the centroid of any set to the centroid of its image.

   Solution to question 4

   If ABCD is a quadrilateral in R², 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 R³ also?

   Solution to question 5

   Prove that any similarity transformation which is not an isometry has a fixed point.

   Solution to question 6

   If AB and A'B' are two line segments in R², prove that there are two similarity transformations mapping A to A' and B to 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.]

   Solution to question 7

   What is the rotational symmetry group of the figure of Exercises 4 Question 6 ?
   [It may help to think of this figure embedded in a cube -- with the edges reaching as far as the faces of the cube. Then look at Exercises 5 Question 3(c).]
   What is the full symmetry group of this figure?

   Solution to question 8