Thus mathematics may be defined as the subject in which
we never know what we are talking about,
nor whether what we are saying is true.
Bertrand Russell
AFFINE AND PROJECTIVE GEOMETRY
Affine Isometries.
1. In the two-dimensional affine euclidian space (E2,(V2,·), ϕ) referred to the orthonormal co-
ordinate system R={O;BV={e1, e2}},we consider the affine transformations:
•the transformation fthat has the following equations with respect to the coordinate
system x0
1= 1 −x1, x0
2= 9 + x2.
•the rotation gof angle 270◦around the point (−1,3).
•the composition h=g◦f.
We ask for:
(a) The matrix form of each one of the transformations.
(b) Check that they are isometries and in this case verify if they are direct or indirect.
(c) Find the invariant points and the invariant subspaces for each one of them.
2. In the three-dimensional space affine euclidian (E3,(V3,·), f ) referred to the orthonormal
coordinate system R={O;BV={e1, e2, e3}} we consider the rotation g:E3→E3with
angle 120◦and axis the line x1= 1 + λ, x2= 2 + λ, x3= 3 + λ. Please obtain:
(a) The matrix equation of gand g3=g◦g◦g:E3→E3.
(b) The subspace of invariant points for gand g3.
3. In the same space of the previous exercise referred to the same coordinate system, obtain:
(a) The matrix form of the symmetry Sπwith respect to the plane π≡x1+x2−2 = 0.
(b) The matrix form of the rotation grwith axis r≡x1−x2= 0
x3= 1.and angle 180◦.
(c) knowing that hk
Cis the homothety over E3with center C(1,1,1) and ratio k=−1,
determine the matrix form of gr◦Sπ◦hk
C:E3→E3,classify it and get the invariant
subspaces.
4. In the same space of the exercise 2 and referred to the same coordinate system, we consider
the line rwith the following parametric equations x1=λ, x2=λ, x3=−1,∀λ∈R.
Please obtain:
(a) The matrix form of the rotation gr:E3→E3,of axis rand angle 180◦.
(b) The matrix form of the symmetry Sπwith respect of the plane πthat is orthogonal to
rand passes through (−1,0,13).
(c) The matrix form of f=Sπ◦gr.Classify it and define the invariant subspaces for f.
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