You are responsible for all of the listed problems, that is, you should be able to do them if asked (say, on a midterm or final exam). Only the starred problems need to be turned in. The others are exercises that are not worth writing out in detail if you understand how to do them.

1.	(complete by 9/2) Review Appendices B, C, and D of the course text.
2.	(complete by 9/9) 1.1-1.5 (that is, Chapter 1, problems 1 through 5)
3.	(due 9/11)* (← i. e. hand these in) Define a distance function D:R2 × R2 → R by the rule D((a,b),(c,d)) = |c-a| + |d-b|. Verify that D satisfies the three properties of a distance function. Determine the following set explicitly: B = { (x,y) | D((x,y),(0,0)) ≤ 1 }.
4.	(9/11)* Give a step-by-step proof that { P + tv | t ∈ R } = { X | X − P ∈ [v] }.
5.	(9/11) 1.9*, 1.10*
6.	(9/18)* Use the inner product description of a line to derive an x-y equation for the line through the point (x0, y0) with normal vector (a,b).
7.	(9/18)* Use the theorem on Existence of Perpendicular Lines to find the line through X=(3,7) perpendicular to the line L = (−1,2) + [ (5,4) ]. Also use the theorem to find the point where the perpendicular line meets L, and the distance d(X,L).
8.	(9/18) 1.13*
9.	(9/18) 1.14* (Hint: Write v = (v1,v2) and w = (w1,w2), and write P − Q = tv + sw as a product of matrices (t,s)D, where D is a 2×2 matrix.)
10.	(9/18)* Prove that if v, w, and h are nonzero vectors with v perpendicular to w and w perpendicular to h, then [v]=[h]. Then prove Theorem 17.
11.	(9/25) Let v ∈ R2 and let T be the function from R2 to R2 defined by TX=X+v. Prove that T is an isometry. Prove that T is a bijection.
12.	(9/25)* Let P ∈ R2 and let H be the function from R2 to R2 defined by HX=2P-X. Prove that H is an isometry. Prove that H is an involution. Prove that H is a bijection.
13.	(9/25)* Let X be a set with a distance function d:X × X → R, and let f and g be isometries of X. Verify that the composition fg is an isometry of X.
14.	(9/25)* Let ΩL be the reflection in a line L. Find two functions f and g from R2 to R2, neither of them the identity function, so that ΩL equals the composition fg. (Hint: There are many ways to do this. One approach involves the use of dilations, such as sending (x,y) to (2x,2y).)
15.	(9/25) 1.22*
16.	(9/25)* Let T(m,n) be the translation given by T(m,n)X = X + (m,n). Let W = { T(m,n) | m and n are integers }, with the operation of composition. Let Z be the { (m,n) | m and n are integers }, a subset of R2. Verify that with the operation of addition, Z is a group (that is, show that it has an identity element and show that each element of Z has an inverse element). Prove that Z is isomorphic to W.
17.	(10/2)* Let L be the line (2,3) + [(1,1)], and let A, B, C, D, and E be the lines perpendicular to L that cross L at the points (8,9), (−2,−1), (3,4), (4,5), and (10,11) respectively. Find a line M perpendicular to L so that OmegaA OmegaB OmegaC = OmegaM. Find a line N perpendicular to L so that OmegaA OmegaB OmegaC OmegaD OmegaE = OmegaN. Find a line F perpendicular to L so that OmegaB OmegaC = OmegaA OmegaF .
18.	(10/2)* Let G be the group of vectors whose entries are integers, that is, {(m,n) | m,n∈Z}, with the operation of vector addition. Let H be the subgroup of elements in G such that n = 0, that is, {(m,0) | m∈Z}. Notice that the coset H + (0,3) looks like {..., (-2,3), (-1,3), (0,3), (1,3), (2,3), ...}. Verify that the cosets H + (0,r) and H + (0,s) are different whenever r is not equal to s. Verify that every element of G is in one of these cosets. Conclude that H has infinite index in G.
19.	(10/2)* Give the details of the calculation that ref(θ)ref(φ) = rot( 2(θ−φ) ).
20.	(10/2) Prove that TR(R2) is isomorphic to the group R2 with the operation of vector addition.
21.	(10/9)* Let REF(0) be the group generated by reflections in lines through the origin, and let ROT(0) be the subgroup consisting of rotations about the origin. Prove that ROT(0) has index 2 in REF(0). First you will want to use a 3 reflections theorem to prove that every element of REF(0) is either a rotation or a reflection.
22.	(10/9)* Let A and B be distinct lines that cross at P. Show that if A is perpendicular to B, then ΩAΩBX = 2P − X for all X, that is, that ΩAΩB equals the half-turn H at P.
23.	(10/9)* Let HP and HQ be the half-turns at the distinct points P and Q. Give two different arguments that HPHQ is a translation along the line L that contains P and Q: (1) Direct calculation, and (2) Start by writing each of HP and HQ as a product of two reflections, one of which is &OmegaL.
24.	(10/9) 1.28*
25.	(10/9) 1.29
26.	(10/30)* Let L be the line [(1,0)], that is, the x-axis in R2. Show that TR(L) is normal in TR(R2), but is not normal in ISOM(R2).
27.	(10/30) Define a function b from R3×R3 to R3 by b(( x1, x2, x3), ( y1, y2, y3)) = x1y1 + x2y2 − x3y3. Check that b is bilinear and symmetric.
28.	(10/30)* For the function b in the previous problem, prove that if b(x,v)=0 for all v in R3, then x=0 (Hint: Since b(x,v) is 0 for all v, it is 0 when v is one of the vectors (1,0,0), (0,1,0), and (0,0,1).)
29.	(10/30)* For the function b in the previous problem, describe the set of all vectors x with b(x,x)=0, the set of all vectors with b(x,x)<0, and the set of all vectors with b(x,x)>0. Find all vectors orthogonal to (1,0,1).
30.	(10/30)* Calculate the determinants of the matrices rot(θ) and ref(θ). Explain geometrically why these are the values one would expect for these determinants.
31.	(10/30)* Let r = (1,x,x2), s = (1,y,y2), and t = (1,z,z2). Let A be the three-by-three matrix whose rows are r, s, and t. Calculate det(A) using our formula. Switch the first two rows and use the formula again to calculate the determinant.
32.	(11/6)* Show that if L is a linear function from R3 to R, then there exists a unique vector z in R3 such that b(z,x)=L(x) for all x in R3 (where b is as in problem 27). Hint: z is similar to (L(e1),L(e2),L(e3)), but a bit different. Figure out what z should be, then follow the proof that we used in class for the case of the standard inner product.
33.	(11/6)* Define the b-cross-product B(u,v) by letting B(u,v) be the unique vector such that b(B(u,v),x)=det(x,u,v) for all x in R3 (we know from the previous problem that there exists a unique such vector). Verify that b( B(u,v), u) = b(B(u,v),v) = 0. Verify that B(u,v) = −B(v,u) and consequently B(u,u)=0. Verify that b(B(u,v),w)=b(u,B(v,w)). Verify that B is bilinear.
34.	(11/6)* For the b-cross-product in the previous problem, calculate a formula for B((u1,u2,u3), (v1,v2,v3)).
35.	(11/6)* The b-cross product satisfies B(B(u,v),w)=b(v,w)u−b(u,w)v (this can be proven analogously to the way we proved the corresponding formula for the regular cross product, but you do not need to carry this out). Use this formula to verify that B(u,v)=0 if and only if u and v are proportional (for the "if" direction, you need the fact from the previous problem that B(u,u)=0).
36.	(11/6)* The b-inner product in R2 is defined by b((x1,x2), (y1,y2))= x1y1−x2y2. We would like to figure out what the "b-cross product" B(u) would be in R2. Use the formula b(B(u),x)=det(x,u) to calculate the formula for B((u1,u2)).
37.	(11/6) Spend some time studying the definition of line and the proof that two non-antipodal points determine a unique line. Draw various pictures of lines and pairs of lines and their poles. Draw pictures illustrating the fact that lines meet perpendicularly exactly when their poles are perpendicular.
38.	(11/13)* Let u and v be unit vectors in R3. Prove that u and v are proportional if and only if u=v or u=−v.
39.	(11/13)* Let u, v, and w be unit vectors in R3. Prove that if w is orthogonal to u and orthogonal to v, then w = (u×v)/|u×v| or w = −(u×v)/|u×v| (first show that (u×v)×w=0, then apply the previous problem to u×v and w).
40.	(11/13) 4.6*, 4.7*
41.	(11/20)* In this problem, we will adapt the calculation of the class example as follows. Use the same line M, with pole (0,1,0), but for L, use the line with pole ( 1/2, 1/root(2), −1/2), where root(2) is the square root of 2. Notice that this new L also meets M at the same point P  = ( 1/root(2), 0, 1/root(2)) (since P is orthogonal to the pole of L and hence lies on L), so we will use the same orthonormal basis {e1, e2, e3} with e3 = P that we used in the class example. Consequently, the matrices for ΩM with respect to these bases will be the same as in the class example, and we do not need to calculate them again. Carry out the following steps: (i) Calculate the matrices of ΩL and of ΩLΩM with respect to the standard basis {E1, E2, E3}. Call the matrix of ΩLΩM A. (As a check on your calculations, the columns of A should form an orthonormal basis.) (ii) Calculate the matrices of ΩL and of ΩLΩM with respect to the orthonormal basis {e1, e2, e3}. Call the matrix of ΩLΩM B. (The columns of B should also form an orthonormal basis.) (iii) Check that Q−1BQ=A, that is, the change of basis matrix Q from {E1, E2, E3}-coordinates to {e1, e2, e3}-coordinates conjugates B to A. (iv) Use the matrix B to work out the rotation angle of ΩLΩM in the (e1,e2)-plane.
42.	(12/4)* Three lines A, B, and C meet at a point P. Let {e1, e2, e3} be an orthonormal basis such that e3 = P. Let S be the circle of radius 1 in the (e1, e2)-plane, and suppose that A meets S in the points e1 and −e1, B meets S in the points e2 and −e2, and C meets S in the points e1/sqrt(2)−e2/sqrt(2) and −e1/sqrt(2)+e2/sqrt(2). By the Three Reflections Theorem, there is a line D through P so that ΩAΩBΩC = ΩD. Find where D meets S, then find a pole of D. (You can calculate the cross-product of vectors expressed in the {e1, e2, e3} basis using the usual formula--- the result will express the cross product in terms of the {e1, e2, e3} basis. Actually, in this example you can figure out a pole of D without doing any additional calculation.)
43.	(12/4)* Find the line D and its pole if in the previous problem we use the line C that meets S in (sqrt(3)/2)e1+(1/2)e2 and (−sqrt(3)/2)e1−(1/2)e2.
44.	(12/4)* Let g be a glide reflection of S2, that is, g = R ΩL where R is a rotation about the pole of L. Check that R ΩL = ΩLR (this is very easy if you work with the matrices of R and ΩL with respect to an orthonormal basis that includes the pole of L).
45.	(12/4) 4.17*, 4.22* (Use an orthonormal basis with e3 equal to the intersection point of alpha and gamma, which is the pole of beta. Write matrices for the three reflections, calculate their product, and see what you get.), 4.23* (First, prove that the formula for HP gives a linear transformation, that is, that HP(aX+bY)=a HP(X)+b HP(Y). That shows that HP can be represented by a matrix. Now, using an orthonormal basis with e3 equal to P, find the matrix for HP, and observe that it is the correct matrix for a half-turn about P.)
46.	4.24 (The antipodal map E is defined by EX=−X. Its matrix with respect to any basis is −I.)
47.	A certain matrix A has first row (x, −2/sqrt(7), sqrt(6)/(3 sqrt(7))), second row (2/3, sqrt(3)/sqrt(7), 2 sqrt(2)/(3 sqrt(7))), and third row (sqrt(2)/3, 0, y). Find x and y so that A is orthogonal. Then, check that AAt = I.