Course MT4521 Geometry and topology

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## Exercises 7

1. The affine span of vectors a1, a2, ... , ar in Rn is the set
{λ1a1 + ... + λrar | λiR, λ1 + ... + λr = 1}.
Prove that it is an affine subspace (i.e. a translated linear subspace).

2. Points a1, a2, ... , ar in Rn are called affinely independent if whenever λ1a1 + ... + λrar = 0 with λ1 + ... + λr = 0 then λ1 = ... = λr = 0.
Prove that for such points the set { a2 - a1, a3 - a1, ... , ar - a1 } is linearly independent.
Prove that for such points the set of vectors { (a1, 1), (a2, 1), ... , (ar , 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.

3. 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.

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

5. 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?

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

7. 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.]

8. 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?

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JOC February 2010