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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 v ( x1 , x2 , , xn ) • Ordered set of numbers: (1,2,3,4) v • Example: (x,y,z) coordinates of pt in If space. n 2 i 1 i x v 1, v is a unit vecto r Indexing into vectors Vector Addition v w ( x1 , x2 ) ( y1 , y2 ) ( x1 y1 , x2 y2 ) v V+w w Scalar Product av a( x1 , x2 ) (ax1 , ax2 ) av v Operations on vectors • sum • max, min, mean, sort, … • Pointwise: .^ Inner (dot) Product v w v.w ( x1 , x2 ).( y1 , y2 ) x1 y1 x2 . y2 The inner product is a SCALAR! v.w ( x1 , x2 ).( y1 , y2 ) || v || || w || cos v.w 0 v w 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 Anm a11 a12 a 21 a22 a31 a32 an1 an 2 a1m a2 m a3m anm Sum: Cnm Anm Bnm cij aij bij A and B must have the same dimensions Matrices Product: Cn p Anm Bm p m cij aik bkj k 1 Identity Matrix: A and B must have compatible dimensions Ann Bnn Bnn Ann 1 0 0 0 1 0 I IA AI A 0 0 1 Matrices Transpose: Cmn A nm cij a ji T If AT A ( A B) A B T T ( AB)T BT AT A is symmetric T 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.