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 command Eg., 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. Indexing into vectors Vector Addition V+w v w Scalar Product av v Operations on vectors • sum • max, min, mean, sort, … • Pointwise: .^ Inner (dot) Product v w The inner product is a SCALAR! 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 Sum: A and B must have the same dimensions Matrices Product: A and B must have compatible dimensions Identity Matrix: Matrices Transpose: If A 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.