Course MT3818 Topics in Geometry

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(Exercises 6)
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(Exercises 8)

Exercises 7

  1. The affine span of vectors a1, a2, ... , ar in Rn is the set
    {lambda1a1 + ... + lambdarar | lambdai belongs R, lambda1 + ... + lambdar = 1}.
    Prove that it is an affine subspace (i.e. a translated linear subspace).

    Solution to question 1

  2. Points a1, a2, ... , ar in Rn are called affinely independent if whenever lambda1a1 + ... + lambdarar = 0 with lambda1 + ... + lambdar = 0 then lambda1 = ... = lambdar = 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 cross 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

  3. Find a group of (n + 1) cross (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

  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 lambda1 + ... + lambdar = 1 then f(lambda1a1 + ... + lambdarar) = lambda1f(a1) + ... + lambdarf(ar). Deduce that an affine map takes the centroid of any set to the centroid of its image.

    Solution to question 4

  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?

    Solution to question 5

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

    Solution to question 6

  7. If AB and A'B' are two line segments in R2, 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.]

    Solution to question 7


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JOC March 2003