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/
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
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
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