Section 7 - Transformations

Question 1

what is a group?

Answer 1

a group is a set together with a binary operation that:

• is closed
• has an identity element
• is associative
• admits inverses

Question 2

for the group of isometries, what is the set?

Answer 2

the set of isometries of the plane

Question 3

for the group of isometries, what is the operation?

Answer 3

function composition

• closed
• identity function is an identity
• function composition is associative
• isometries are 1-1, onto so if f: ℝ²→ℝ² is an isometry, so is f⁻¹ (which exists), and f◦f⁻¹ = f⁻¹◦f = identity function

Question 4

a transformation (f: ℝ²→ℝ²) preserves vector addition & scalar mult. iff …

Answer 4

f(u+v) = f(u) + f(v) and f(ɑu) = ɑf(u)

this transformation is linear since it maps lines to lines

Question 5

prove that if f: ℝ²→ℝ² is a linear transformation, then f maps lines to lines

Answer 5

• let a + ɑb, ɑ∈ℝ, be a line in ℝ²

- its image is f(a + ɑb) = f(a) + f(ɑb) = f(a) + ɑf(b), which is a line

Question 6

prove that parallel lines map to parallel lines

Answer 6

• let l₁ = a₁ + ɑb & l₂ = a₂ +βb, ɑ,β∈ℝ, be lines in ℝ²
• then l₁//l₂
• then f(l₁) = f(a₁) + ɑf(b) & f(l₂) = f(a₂) +βf(b)
• so f(l₁)//f(l₂)

Question 7

what matrix do we use to represent f: ℝ²→ℝ², a linear transformation?

Answer 7

⎡a   b⎤
⎣c   d⎦
=M
then f((x,y)) is the matrix product:
⎡ax + by⎤
⎣cx + dy⎦

Question 8

if f₁ = (a₁x + b₁y , c₁x + d₁y) & f₂ = (a₂x + b₂y , c₂x + d₂y), how do we calculate f₁◦f₂((x,y)) using matrices?

Answer 8

first:
⎡a₂   b₂⎤⎡x⎤
⎣c₂   d₂⎦⎣y⎦
then the matrix from above by:
⎡a₁   b₁⎤
⎣c₁   d₁⎦

in summary:
⎡a₁ b₁⎤⎡a₂ b₂⎤⎡x⎤
⎣c₁ d₁⎦⎣c₂ d₂⎦⎣y⎦

Question 9

our square matrix M is invertible iff….

Answer 9

…. there exists a matrix M⁻¹ s.t. MM⁻¹ = M⁻¹M = 𝙸
&
…. det(M)≠0

Question 10

if ad-bc≠0 then M⁻¹ = ?

Answer 10

⎡d -b⎤(1/ad-bc) = M⁻¹

⎣-c a⎦

Question 11

how do we know any 2x2 matrix M represents a linear transformation?

Answer 11

M(u+v) = Mu + Mv
M(ɑu) = ɑMu

Question 12

what is the matrix that represents a rotation by θ

Answer 12

⎡cosθ -sinθ⎤
⎣sinθ cosθ⎦
det = 1 so invertible

its inverse is:
⎡cos(-θ) -sin(-θ)⎤
⎣sin(-θ) cos(-θ)⎦

Question 13

what is the matrix that represents reflection about the x axis

Answer 13

⎡1 0⎤
⎣0 -1⎦
inverse is itself

Question 14

what is the matrix that represents stretch by a factor k(≠0) in x direction

Answer 14

⎡k 0⎤
⎣0 1⎦
det = k ≠ 0 so invertible

Question 15

any invertible linear transformation of ℝ² is a composition of…

Answer 15

• reflections about lines through O (recall that to reflect about lines through O: (1) rotate, (2) reflect about x axis, (3) rotate back)
• stretches in the x direction by non-zero factors of k

Question 16

what is an affine transformation of the plane?

Answer 16

a function g:ℝ²→ℝ² such that for all ū∈ℝ²:

g(ū) = Mū + c, c∈ℝ², where M is an invertible linear transformation

Question 17

what do affine transformations preserve?

Answer 17

-lines
-parallels
they do not preserve position

Question 18

what is affine geometry concerned with?

Answer 18

the things that are preserved by affine transformations (i.e. lines & //s)

Question 19

in ℝℙ¹ what is a linear fractional that sends the line s (y=(1/s)x) to f(s)

Answer 19

f(s) = (as+b)/(cs+d)

Question 20

what is the 2-sphere?

Answer 20

the unit sphere in ℝ³

{(x,y,z) : x²+y²+z² = 1}

Question 21

what is the notation for the 2-sphere?

Answer 21

𝒮²

Question 22

what are the equators of 𝒮²

Answer 22

intersections of 𝒮² & planes through the origin in ℝ³

Question 23

what are the isometries of 𝒮²

Answer 23

• rotation
• reflection about a plane through O in ℝ³ (reflection about an equator)
• any composition of isometries
• antipodal map: f((x,y,z)) = (-x,-y,-z)

Question 24

what is the antipodal map of 𝒮²

Answer 24

composition of three reflections:

• reflect in yz plane: (x,y,z) → (-x,y,z)
• reflect in xz plane: (-x,y,z) → (-x,-y,z)
• reflect in xy plane: (-x,-y,z) → (-x,-y,-z)

Question 25

how de we prove that if f:ℝ³→ℝ³ is an isometry s.t. f(0)=0, then the restriction of f to 𝒮² is an isometry of 𝒮²

Answer 25

• f must map the set of pts at distance 1 from 0 to itself, and preserves distance between them
• thus it is an isometry of 𝒮²

the converse of this statement is also true

Question 26

state the three reflections thm for 𝒮²

Answer 26

any isometry of 𝒮² is a composition of 1, 2, or 3 reflections

(we also have that any isometry of 𝒮² is uniquely determined by the images of points A,B,C not on the same equator (great circle))

Question 27

state the rotation group of the sphere

Answer 27

• isometries of 𝒮²

- function composition

Question 28

let r₁&r₂ be isometries of 𝒮² determined by reflection about planes P₁&P₂ through O. Then r₂◦r₁ is……

Answer 28

… a rotation of 𝒮² about the line of intersection of P₁&P₂ by 2θ, where θ is the angle from P₁ to P₂

Question 29

do rotations of 𝒮² commute?

Answer 29

no

Question 30

the composition of any even number of reflections of 𝒮² is a ……

Answer 30

rotation