Exercises and Exams

Question 1

1.1 a)
Find matrix corresponding to reflection in y=ax.

Question 2

1.1 b)
Find A, b such that T(x) = Ax + b is reflection in y=ax + b

Question 3

1.1 e)
is the composition of two reflections always a rotation?

Question 4

1.2 b)
find the rotation that maps (0,0) -> (2,2), (1.-1) -> (1,1).`

Question 5

1.2 d)
can you tell whether a matrix is a reflection or a rotation from its eigenvalues.

Question 6

1.4
Show that if (X, d_X) and (Y, d_Y) are isometric Euclidean spaces, then they are both isometric to E^n for some n.

Question 7

1.5
Show that if x,y,z colinear, then z = x + \lambda(y-x), \lambda \in (0,1).

Question 8

3.1
T: R^3 -> R^3 is an isometry such that T(0,1,1) = (2,1,1), T(0,1,-1) = (2,1,-1), T(1,-1,0) = (0,2,0), T(-1, -1, 0) = (0,0,0).

a) What is T(0)?

b) find A, b such that T(x) = Ax + b. Are they unique?

Question 9

3.2
Prove that for c \in R^n, not zero, and b \in R, {v \in R^n : <v, c> = b} is a hyperplane.

Question 10

3.3
Prove that if R is a reflection, and T is any isometry, then T^-1(R(T))) is a reflection.

Question 11

3.5
Suppose R reflects \Pi = V + b. Show that R commutes with x ._ x + v.

Question 12

3.9
If L = P + Rv, L’ = P’ + Rv’, then L=L’ iff. v’, P-P’ \in Rv.

Question 13

4.2.ii)
Let L:R^(n+1) -> R^(n+1) be a linear isometry, show that x -> L(x) sends great circles to great circles on S^2

Question 14

6.1

Find a,b such that L_1 = H^2 \cap {x_1 = ax_2 + bx_2} projects to line in (x2-x3)-plane that contains point (2,0).

6.2
Find matrix A \in O(1,2) such that L1 is mapped to x_2 = 0.

6.3
Is the above A unique.

Question 15

7.1.i)
Show that x x_L y is Lorentz orthogonal to x and y.

Question 16

8.1
Find number of planes in P^3(F_5)

Question 17

8.2a)
Write down all projective linear maps \phi:P^1 -> P^1 such that (1:0) -> (1:0) and (0:1) -> (0:1)

8.2 b)
find unique A \in PGL such that (1:0) -> (1:0), (0:1) -> (0:1) and (1:1) -> (1:2).

Question 18

A2.1.c)
if A is a rotation 90 degrees about x-axis, and B is 90 degress about y-axis, what is AB.

Question 19

A3.2
if \Pi = {x_1 = ax_2 + bx_3}, and L = H^2 \cap \Pi,.

a) what is a condition on a,b such that L is non-empty.

b) suppose b=0, find equatioon of L in (x_2,x_3)-plane,

e) Let L_1 project to x_2=x_3 and L_2 project to x_2 = -x_3. give equatin of plane such that it intersects neitehr L1 or L2.

Question 20

23.1.iv)
If T is a non-trivial isometry of R^3 and \Pi is a plane in R^3 such that T(\Pi) = \Pi, then T is a reflection in \Pi, true or false?

Question 21

23.1.v)
every isometry of R^3 is the composition of at most 2 reflections. true or false.

Question 22

23.1.xii)
There exists projective transofrmation such that (1:0) -> (1:1)
(1:3) ->(-2:1),
(1:1) -> (1:0),
(1:2) -> (-3:1)

Question 23

23.2.c)
Prove that if A \in R^3x3 and T_A : H^2 -> H^2 is an isometry, than A \in O(1,2).

Question 24

23.2.e)
Prove the triangle inequality for the hyperbolic metric on H^2.

Question 25

23.3.b)
Prove that if \Pi is a plane in P^3 and L is a line, then \Pi and L must have at least one point in common.

Question 26

23.3.c)
Give an example of na plane and line in P^4 such that \Pi \cap L = \varnothing.

Question 27

A4.1
Compute number of points and lines in P^1(F_5).

Question 27

A4.2
Which of the following sets of points are colinear in P^2:
A = {(2:3:1), (1:3:2), (2:4:2)}
B = {(1:2:3), (3:2:1), (2:4:2)}.
Give equation of plane.

Question 28

22.1.f)
Show that two distinct projective lines in P^2 intersect at exactly one point.

Question 29

22.3 c)
Which hyperbolic lines of H^2 intersect?
\Pi_1 = {x_2 + x_3 = 0}
\Pi_2 = {x_1 + x_2 - 3x_3 = 0}
\Pi_3 = {x_3 = 0}

Question 30

21.1.c)
Let n>= 1 and L: R^n -> R^n be a linaer motion with real eigenvalue. then is \lambda = \pm 1?

Question 31

21.1.f)
For every n >= 2, is it true that every Euclidean line L in R^n such that P \notin L there exists L’ st P \in L’ and L \cap L’ = \varnothing?

Question 32

21.1.g)
for every n>=2, is it true that every great cirlce C in S^n such that P \notin C, there exists great circle C’ such that P \in C’ and C \cap C’ = \varnothing?

Question 33

21.1.h)
for every n += 2, is it true that for every hyperbolic line L in H^2, and P \notin L, there eixsts hyperbolic line L’ such that P \in L’ and L \cap L’ = \varnothing?