Announcements Problem Set 2, handed out today, due next Tuesday. Late Homework should be turned into my office with date and time written on it. Mail problem.

Announcements

• Problem Set 2, handed out today, due next Tuesday.

• Late Homework should be turned into my office with date and time written on it.

• Mail problem sets in one zipped file to farrell (at) cs.umd.edu. When emailing matlab code to Ryan, use subject: CMSC 426 Matlab code.

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.

Starting Matlab

• For PCs, Matlab should be a program.

• For Sun’s:

Numerical

Analysis and Visualization

Matlab 6.1

Help

• help• help commandEg., help plus• Help on toolbar• demo• Tutorial:

http://amath.colorado.edu/scico/tutorials/matlab/

Matlab interpreter

• Many common functions: see help ops

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

Indexing into vectors

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.

How do we prove these properties of the inner product? Let’s start with the fact that orthogonal vectors have 0 inner product. Suppose one vector is (x,y), and WLOG x,y>0. Then, if we rotate that by 90 degrees counterclockwise, we’ll get (y, -x). Rotating the vector is just like rotating the coordinate system in the opposite direction. And (x,y)*(y,-x) = xy – yx = 0.

Next, note that v*w = (v*w)/(||v||||w||) * ||v||||w|| This means that if we can show that when v and w are unit vectors v*w = cos alpha, then it will follow that in general v*w = ||v|| ||w|| cos alpha. So suppose v and w are unit vectors.

Next, note that if w1 + w2 = w, then v*w = v*(w1+w2) = v*w1 + v*w2. For any w, we can write it as the sum of w1+w2, where w1 is perpendicular to v, and w2 is in the same direction as v. So v*w1 = 0. v*w2 = ||w2||, since v*w2/||w2|| = 1. Then, if we just draw a picture, we can see that cos alpha = ||w2|| = v*w2 = v*w.

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

Indexing into matrices

Files

Matlab

Functions

• Format: function o = test(x,y)

• Name function and file the same.

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

• Look at sample function

Images

• Black and white image is a 2D matrix.

• Intensities represented as pixels.

• Color images are 3D matrix, RBG.

• Matlab

Debugging

• Add print statements to function by leaving off ;

• keyboard

• debug and breakpoint

Conclusions

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

• The inner product will be very important.