what is a group?
a group is a set together with a binary operation that:
- is closed
- has an identity element
- is associative
- admits inverses
for the group of isometries, what is the set?
the set of isometries of the plane
for the group of isometries, what is the operation?
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
a transformation (f: ℝ²→ℝ²) preserves vector addition & scalar mult. iff …
f(u+v) = f(u) + f(v) and f(ɑu) = ɑf(u)
this transformation is linear since it maps lines to lines
prove that if f: ℝ²→ℝ² is a linear transformation, then f maps lines to lines
- 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
prove that parallel lines map to parallel lines
- 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)
what matrix do we use to represent f: ℝ²→ℝ², a linear transformation?
⎡a b⎤ ⎣c d⎦ =M then f((x,y)) is the matrix product: ⎡ax + by⎤ ⎣cx + dy⎦
if f₁ = (a₁x + b₁y , c₁x + d₁y) & f₂ = (a₂x + b₂y , c₂x + d₂y), how do we calculate f₁◦f₂((x,y)) using matrices?
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⎦
our square matrix M is invertible iff….
…. there exists a matrix M⁻¹ s.t. MM⁻¹ = M⁻¹M = 𝙸
&
…. det(M)≠0
if ad-bc≠0 then M⁻¹ = ?
⎡d -b⎤(1/ad-bc) = M⁻¹
⎣-c a⎦
how do we know any 2x2 matrix M represents a linear transformation?
M(u+v) = Mu + Mv M(ɑu) = ɑMu
what is the matrix that represents a rotation by θ
⎡cosθ -sinθ⎤
⎣sinθ cosθ⎦
det = 1 so invertible
its inverse is:
⎡cos(-θ) -sin(-θ)⎤
⎣sin(-θ) cos(-θ)⎦
what is the matrix that represents reflection about the x axis
⎡1 0⎤
⎣0 -1⎦
inverse is itself
what is the matrix that represents stretch by a factor k(≠0) in x direction
⎡k 0⎤
⎣0 1⎦
det = k ≠ 0 so invertible
any invertible linear transformation of ℝ² is a composition of…
- 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
what is an affine transformation of the plane?
a function g:ℝ²→ℝ² such that for all ū∈ℝ²:
g(ū) = Mū + c, c∈ℝ², where M is an invertible linear transformation
what do affine transformations preserve?
-lines
-parallels
they do not preserve position
what is affine geometry concerned with?
the things that are preserved by affine transformations (i.e. lines & //s)
in ℝℙ¹ what is a linear fractional that sends the line s (y=(1/s)x) to f(s)
f(s) = (as+b)/(cs+d)
what is the 2-sphere?
the unit sphere in ℝ³
{(x,y,z) : x²+y²+z² = 1}
what is the notation for the 2-sphere?
𝒮²
what are the equators of 𝒮²
intersections of 𝒮² & planes through the origin in ℝ³
what are the isometries of 𝒮²
- 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)
what is the antipodal map of 𝒮²
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)
