# Math 331 - Transformation Geometry: Syllabus and Homework

Annuncements:

• Due to the missed classes, the two tests are pushed back by one week, so they will be on March 2 and April 13.
• For reflections in R^n, I refer to the following source .

Textbook for assignments: George Martin, Transformation Geometry. An Introduction to Symmetry

All problems are worth 10 points unless otherwise specified.

• Due on 2/9
• Let f:R^3->R^2 and g:R^2->R^3 be given by f(e1)=e1+e2, f(e2)=2e1-e2, f(e3)=e2; g(e1)=e1-e2+e3, g(e2)=e1+3e2-2e3. Compute (fog)(3e1+2e2).
• Show that, given a linear map f, we have: (1) f is an isometry if and only if (2) f preserves the Euclidean inner product, that is, (f(x),f(y))=(x,y). You need to prove both implications. For (1) implies (2), start by rewriting the isometry property in terms of the scalar product, and then use the bilinearity of the latter. For (2) implies (1), replace both x and y with x-y in (2).
• Give an example of: 1) an isometry which is not a linear map; 2) a linear map which is not an isometry; 3) an affine map which is neither an isometry nor a linear map (you might combine the previous two examples). It is easiest to think of maps on the plane R^2-->R^2, and write down the formulas describing them. Justify your choices.
• Due on 2/23
• Prove that translations and halfturns map any line {x+tv : t in R} to a parallel line (same direction v).
• page 21/3.10
• Due on 3/2
• page 40/5.1 (read the text at the bottom of page 35 and top of 36).
• Use a reflection to construct (with the ruler and compass) a quadrilateral given all the side lengths (a,b,c,d) and the fact that a given diagonal bisects an angle (say the one formed by the sides a and b).
• Proofs for the test (from the Steinberger lecture notes on the web): Lemma 2.4.3 and Corollary 2.4.4, Proposition 2.5.1, Theorem 4.1.5. Other problems on test based on: compositions of translations and halfturns (see page 20-21/3.2 -- in class, 3.10 -- homework), formulas for various transformations (examples in class), geometric constructions based on reflections (examples in class and homework), compositions of reflections (examples in class and homework).
• Due on 3/23 page 41/5.9
• Due on 3/30 page 50-51/6.3, 6.16
• Due on 4/6 page 61/7.10, page 68/8.2, page 76/9.8.
• Due on 4/13 page 146/13.27
• Optional practice problems (some were solved in class): page 60-61/7.1--7.14, 7.17; page 68-70/8.1--8.4, 8.7c--h, 8.9--8.11, 8.13, 8.14; page 76-77/9.1--9.12.