Matlab tutorial and Linear Algebra Review Today’s goals: Learn enough matlab to get started. Review some basics of Linear Algebra Essential for geometry.

Matlab tutorial and Linear Algebra Review

• Today’s goals:• Learn enough matlab to get started.• Review some basics of Linear Algebra• Essential for geometry of points and lines.• But also, all math is linear algebra.• (ok slight exaggeration).• Many slides today adapted from Octavia

Camps, Penn State.

Vectors

• Ordered set of numbers: (1,2,3,4)

• Example: (x,y,z) coordinates of pt in space. runit vecto a is ,1 If

),(

1

2

,,21

vv

xv

xxxvn

i i

n

Vector Addition

),(),(),( 22112121 yxyxyyxx wv

vvww

V+wV+w

Scalar Product

),(),( 2121 axaxxxaa v

vv

avav

Operations on vectors

• sum

• max, min, mean, sort, …

• Pointwise: .^

Inner (dot) Product

vv

ww

22112121 .),).(,(. yxyxyyxxwv

The inner product is a The inner product is a SCALAR!SCALAR!

cos||||||||),).(,(. 2121 wvyyxxwv

wvwv 0.

Matrices

nmnn

m

m

m

mn

aaa

aaa

aaa

aaa

A

21

33231

22221

11211

mnmnmn BAC Sum:Sum:

ijijij bac

A and B must have the same A and B must have the same dimensionsdimensions

Matrices

pmmnpn BAC Product:Product:

m

kkjikij bac

1

A and B must have A and B must have compatible dimensionscompatible dimensions

nnnnnnnn ABBA

Identity Matrix:

AAIIAI

100

010

001

Matrices

mnT

nm AC Transpose:Transpose:

jiij ac TTT ABAB )(

TTT BABA )(

IfIf AAT A is symmetricA is symmetric

Matrices

Determinant:Determinant: A must be squareA must be square

3231

222113

3331

232112

3332

232211

333231

232221

131211

detaa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

122122112221

1211

2221

1211det aaaaaa

aa

aa

aa

Matrices

IAAAA nnnnnnnn

11

Inverse:Inverse: A must be squareA must be square

1121

1222

12212211

1

2221

1211 1

aa

aa

aaaaaa

aa

Indexing into matrices

Euclidean transformations

2D Translation

tt

PP

P’P’

2D Translation Equation

PP

xx

yy

ttxx

ttyy

P’P’tt

tPP ),(' yx tytx

),(

),(

yx tt

yx

t

P

2D Translation using Matrices

PP

xx

yy

ttxx

ttyy

P’P’tt

),(

),(

yx tt

yx

t

P

1

1

0

0

1' y

x

t

t

ty

tx

y

x

y

xP

tt PP

Scaling

PP

P’P’

Scaling Equation

PP

xx

yy

s.xs.x

P’P’s.ys.y

),('

),(

sysx

yx

P

P

PP s'

y

x

s

s

sy

sx

0

0'P

SPSP '

Rotation

PP

PP’’

Rotation Equations

Counter-clockwise rotation by an angle Counter-clockwise rotation by an angle

y

x

y

x

cossin

sincos

'

'

PP

xx

Y’Y’PP’’

X’X’

yy R.PP'

Degrees of Freedom

R is 2x2 R is 2x2 4 elements4 elements

BUT! There is only 1 degree of freedom: BUT! There is only 1 degree of freedom:

1)det(

R

IRRRR TT

The 4 elements must satisfy the following constraints:The 4 elements must satisfy the following constraints:

y

x

y

x

cossin

sincos

'

'

Stretching Equation

PP

xx

yy

SSxx.x.x

P’P’SSyy.y.y

y

xs

s

ys

xs

y

x

y

x

0

0'P

),('

),(

ysxs

yx

yx

P

P

S

PSP '

Stretching = tilting and projecting(with weak perspective)

y

xs

s

sy

xs

s

ys

xsy

x

yy

x

y

x

10

00

0'P

Linear Transformation

y

xs

s

s

y

xs

s

y

x

dc

ba

y

x

y

y

x

sincos

cossin

10

0

sincos

cossin

sincos

cossin0

0

sincos

cossin

'PSVD

Affine Transformation

1

' y

x

tydc

txbaP

Files

Functions

• Format: function o = test(x,y)

• Name function and file the same.

• Only first function in file is visible outside the file.

Conclusions

• Quick tour of matlab, you should teach yourself the rest. We’ll give hints in problem sets.

• Linear algebra allows geometric manipulation of points.

• Learn to love SVD.