• Line of Best Fit• Geometry of Linear Maps• Markov Chains• Orthonormal Matrices

3. Topics

Line of Best Fit

Over-determined systems,

e.g., coin flipping:

30 16

60 34

90 51

m

m

m

Number of flips 30 60 90

Number of heads 16 34 51

( m = fraction of heads )

There is no solution since

16 30

34 60

51 90

m m

R

Best fit is

30

60

90

16 30

34 60

51 90

proj m

16 30

34 6030

51 9060

30 3090

60 60

90 90

7110

12600m 0.56

Circulation duration of U.S. currency:

denomination 1 5 10 20 50 100

average life (yrs) 1.5 2 3 5 9 20

1 1

1 5

1 10

1 20

1 50

1 100

A

life = m denom + b

A x = vb

m

x

1.5

2

3

5

9

20

v

1T T A A A v

Best fit is projection of v onto column space of A :

1.05

0.18

bproj

m

A v

Fitting curve a f(x) + b g(x) to a set of data points { m(x1), …, m(xn) }

1 1

n n

f x g x

f x g x

A A x = ma

b

x

1

n

m x

m x

m

1T T A A A v

Best fit is projection of m onto column space of A :

aproj

b

A m

Exercises 3.Topics.1.

1. When the space shuttle Challenger exploded in 1986, one of the criticisms made of NASA’s decision to launch was in the way the analysis of number of O-ring failures versus temperature was made (of course, O-ring failure caused the explosion). Four O-ring failures will cause the rocket to explode. NASA had data from 24 previous flights.

The temperature that day was forecast to be 31ºF.(a) NASA based the decision to launch partially on a chart showing only theflights that had at least one O-ring failure. Find the line that best fits theseseven flights. On the basis of this data, predict the number of O-ring failureswhen the temperature is 31, and when the number of failures will exceed four.(b) Find the line that best fits all 24 flights. On the basis of this extra data,predict the number of O-ring failures when the temperature is 31, and when the number of failures will exceed four.Which do you think is the more accurate method of predicting?

Geometry of Linear Maps

1xf x e 2

2f x x 1 2h x x 2h x x

Linear MapsNon-Linear Maps

Rotation: Linear Projection: Linear

Every H can be written as H = P B Q, where B is a partial identity matrix,and P , Q are products of elementary matrices, Mi (k), Pi , j , and Ci , j (k).

B ~ projection.

Mi (k) ~ dilation

Pi , j ~ reflection

Ci , j (k) ~ skew

Linear map h maps subspaces into subspaces. E.g., line through 0 to line through 0.Dim h(V) cannot be greater than Dim V → a line can’t map onto a plane.

Calculus: Near x0 , f(x) f(x0) + (xx0) f (x0) is a linear map.

Multi-dim: f : n → m , f(x) f(x0) + [ (xx0) · ] f (x0)

Chain rule:

d g fx g f x f x

dx

f dilates ( x ) by a factor of f (x).

g dilates ( f (x) ) by a factor of g (f (x)).

g f dilates ( x ) by a factor of g (f (x)) f (x).

Multi-dim: proj, shear, etc…

Exercises 3.Topics.2.

1. What combination of dilations, flips, skews, and projections produces the map h: 3 → 3 represented with respect to the standard bases by this matrix?

1 2 1

3 6 0

1 2 2

Markov Chains

Player starts with 3 dollars & bets a dollar for each coin toss.

Game is over when he has no money or up to 5 dollars.

6 possible states: s0 , s1 , s2 , s3 , s4 , s5 with game over whenever s0 or s5 is reached.

Let pi (n) be the probability for him to be in state si after n tosses. Then

0 0

1 1

2 2

3 3

4 4

5 5

1 1 .5 0 0 0 0

1 0 0 .5 0 0 0

1 0 .5 0 .5 0 0

1 0 0 .5 0 .5 0

1 0 0 0 .5 0 0

1 0 0 0 0 .5 1

p n p n

p n p n

p n p n

p n p n

p n p n

p n p n

1 11 0.5 0.5i i ip n p n p n

1 21 0.5p n p n 0 0 11 0.5p n p n p n

4 31 0.5p n p n 5 5 41 0.5p n p n p n

i = 2, 3

mi j = probability of state j changing to state i

1i ji

m

Starting with 3 dollars:

Probability for game to be over is .125+.375 = .5 at n = 4,

and .396 + .59676 = .99276 at n = 24.

The coin toss is a Markov process / chain (no memory effect).

States 0 & 5 are absorptive.

M is the transition matrix.

p is the probability vector.

Exercises 3.Topics.3

.

Orthonormal Matrices

Euclidean geometry:

2 figures are the same (congruent) if they have the same size & shape.

f : 2 → 2 is an isometry if it preserves distances.2 figures are congruent if they are related by an isometry.The followings are preserved under isometry:• Collinearity.• Between-ness of points.• Properties of a triangle.• Properties of a circle.

Klein’s Erlanger Program proposes that each kind of geometry — Euclidean, projective, etc.— can be described as the study of the properties that are invariant under some group of transformations.

Characterization of Isometries by Means of Linear Algebra

The only non-linear isometries is translation.1 1

0

x x x

y y y

If f is an isometry that sends 0 to v0 , then v f(v) v0 is linear.

A linear transformation t of the plane is an isometry iff

1 2 1t t e e and 1 2 0t t e e

Proof : From Pythagorean theorem.

Proof : Direct calculation.

Matrix representation of a linear isometry is orthonormal,

i.e., its columns are mutually orthogonal & of length one.

Note: most people call such matrices orthogonal & define it by M MT = I.

Rotation

Reflection

Euclidean study of congruence:(i) a rotation followed by a translation, or (ii) a reflection followed by a translation ( glide reflection).

2 Figures are similar if they are congruent after a change of scale.

i.e., if there exists an orthonormal matrix T s.t. points q & p on them are related by

0k q T p p

Exercises 3.Topics.4.

1. Write down the formula for each of these distance-preserving maps.(a) the map that rotates π/ 6 radians, and then translates by e2.(b) the map that reflects about the line y = 2x.(c) the map that reflects about y =2x and translates over 1 and up 1.