Vector Spaces. 1 Theory. 2 Matlab. Rules and Operations

Transcription

1 Vector Spaces Rules and Operations 1 Theory Recall that we can specify a point in the plane by an ordered pair of numbers or coordinates (x, y) and a point in space with an order triple of numbers (x, y, z). This also defines a vector with base at the origin and terminal point at the specified point in the plane or in space. In general, an n-dimensional real vector is an ordered n-tuple of numbers (x 1, x 2..., x n ) and is a point in R n. If u and v are two vectors positioned so that the initial point of v is at the terminal point of u, then the vector that is the sum u + v, is the vector from the initial point of u to the terminal point of v. If v is a vectors and c is a scalar, then the scalar multiple cv is a vector whose magnitude is c times the magnitude of v pointing in the same direction as v if c > 0 and in the opposite direction of v if c < 0. If c = 0 or v = (0, 0,..., 0) then cv = (0, 0,..., 0). The difference v u is v+ 1u and is a vector pointing from the terminal point of u to the terminal point of v The centroid of a collection of points in R n can be thought of as the collection s center of mass. It is the point whose ith component is the average value of the ith components of all the points in the collection. 2 Matlab Useful Matlab Functions/Concepts: Manipulating Vectors A k-dimensional real vector is stored in Matlab as an ordered list of real numbers. Vectors can be written as row vectors or column vectors. When defining vectors the coordinates in a row vector are separated by commas and coordinates in a column vector are separated by semicolons. We have to be careful when trying to add vectors: row(column) vectors can only be added to row(column) vectors and they must be the same dimension. 1

2 >> rowvector1 = [1, 2, 3]; >> rowvector2 = [4, 2, -1]; >> columnvector1 = [1 ; 2 ; 3]; >> columnvector2 = [1 ; 2 ; 3 ; 4]; >> rowvector1+rowvector2 >> rowvector1+columnvector1 >> columnvector1+columnvector2 (Hint: The last two commands should give you errors.) Scalar multiplication is achieved using the * symbol with a scalar and a vector. >> vector = [1, 2, 3]; >> scaledvector = 5*rowvector1 The dot product of two vectors is also achieved with the * symbol but you must be careful that the first term is a row vector and the second term is a column vector. >> rowvector1 = [1, 2, 3]; >> columnvector1 = [1 ; 2 ; 3]; >> rowvector1*columnvector1 Note that >> rowvector1*rowvector1 will produce an error and the operation >> columnvector1*rowvector1 is beyond the scope of this course. Structures Matlab can store many different types of data. Integers, real numbers, and strings are among the most commonly used. However, a table in Matlab can only contain data of one type. Often we may wish to group data of different types. For example, if our data is patient medical information we may want to record the patients name (a string), their age (an integer), and their weight and height (positive real numbers). One way to do this in Matlab is with a data type called a 2

3 structure. A structure is like a database of patient information where each patient gets an entry in the database and each entry can contain fields with data of different types. If the variable patients is a structure in Matlab, the syntax for accessing the value in the ith entry of a structure is patients(i). This will return all of the fields in that entry. If one of those fields is given the variable name age, the syntax for accessing that particular field for the ith entry is patients(i).age. >> patients = struct( name,[], age,[],); >> patients(1).name = John Smith ; >> patients(1).age = 47; >> patients(1).weight = 198.1; >> patients(1).height = 6.1; >> patients(2).name = Jane Doe ; >> patients(2).age = 26; >> patients(2).weight = 109.6; >> patients(2).height = 5.5; rand(rows,columns) There are many reasons one might want to generate random numbers. Matlab can produce a table with r rows and c columns with real numbers entries randomly chosen between 0 and 1 using the command >> rand(r,c). So rand(1,3) can be used to generate a random RGB color. This can also be used to generate a random number between a and b by >> a+(b-a)*rand(r,c). >> minvalue = -3; >> maxvalue = 17.2; >> rows = 5; >> cols = 8; >> randoms = minvalue + (maxvalue-minvalue)*rand(rows,cols); scatter(x,y,size,color) scatter(x,y,size,color) performs the same same function as scatter3(x,y,size,color) but it takes two dimensions of data instead of three. 3

4 hold You may have reason to plot several plots on the same figure. To do so you must toggle on the command hold after the first figure is plotted, otherwise Matlab will overwrite the existing plot. Alternatively you can open a blank figure window with the command figure and toggle on the command hold after that blank figure is made. All subsequent plots will be put into the same figure window until hold is toggled off. >> points1 = rand(50,2); >> points2 = rand(50,2)-1; >> figure >> hold >> scatter(points1(:,1),points1(:,2),10,[1,0,0]) >> scatter(points2(:,1),points2(:,2),5,rand(1,3)) 3 Experimentation A program called dotproduct.m can be found on the course website. Download it and place it in your Matlab working directory. This function takes in two real (equal length) vectors in R 2 or R 3 and plots the vectors in the plane or in space. This program also outputs the computation of the dot product of these two vectors to the command window. >> v = [1,0,-1]; >> w = [sqrt(3),sqrt(3),sqrt(3)]; >> d = dotproduct(v,w); After running the above example, try a few other input vectors in both two and three dimensions to familiarize yourself with the program, then proceed with the investigations below. 4

5 3.1 Investigation 1 Step 1: Define the parameter c = 1; and vectors v = c*[1,2,3]; and w = [2,0,0];. Notice that the vector v depends on what you chose for the parameter c. Step 2: Run dotproduct(v,w) for several choices of c. What do you observe about the relationship between c and the dot product? Does the dot product appear to depend on the magnitude of the vectors? Step 3: Run dotproduct(v,w) for several choices of unit vectors. Does the dot product appear to depend on the angle between the vectors? 3.2 Investigation 2: Step 1: Define the parameter theta = 0; and vectors v = [cos(theta),sin(theta)]; and w = [1,0];. Notice that the vector v depends on what you chose for the parameter theta. Step 2: Run dotproduct(v,w) for several choices of theta. What do you observe about the relationship between theta and v? What do you observe about the relationship between theta and the dot product? Can the dot product be made positive? Negative? Zero? If so, what can you say about the relationship between the vectors v and w in each case? 3.3 Bonus: Step 1: Fix w = [1 ; 0] and define v = [cos(theta),sin(theta)]. Step 2: Using a for loop, compute the dot product of w with v for many choices of theta in the range 0 to π and store these in a list. 5

6 Step 3: Plot the list of dot product values against the list of different theta values. Do you recognize the function which you plotted? Conjecture an answer to the following question: How does the dot product between two unit vectors depend on the angle between them? 4 Application The antigenic binding data for 79 different antisera was collected for 273 influenza viruses between 1968 and Building on the ideas in the previous lab, we think of this data as 273 points in 79-dimensional antigenic space. Since we cannot visualize such a high dimensional space, a complicated projection onto a plane was carried out. The resulting data can be thought of as a collection of 273 points in R 2 and can be used to visualize the antigenic evolution of influenza between 1968 and This data has been provided for you in a file on the course website called antigenicdata.mat. Download the data file antigenicdata.mat from the course website and load it into Matlab. This will load a structure, antigenicdata, which contains 273 entries, each of which contains the fields id (a unique string identifying the virus), coords (a 2-dimensional row vector), and strain (an integer between 1 and 11 identifying a strain). Note: Strains are listed in chronological order over the time period 1968 to Create a new Matlab script called lastname operations.m which performs the following steps and use it to answer the questions in this lab. It may be useful to use the following command throughout your code: >> strain = [antigenicdata([antigenicdata.strain]==i).coords]; This is an example of a powerful technique called logical indexing. This stores a list of the xy-coordinates of only those entries in the structure whose strain value is equal to i. In this way you can easily access only the strain you are interested in. 4.1 Coding: Step 1: Scatter plot the planar antigenic data and color code the viruses according to strain. 6

7 Step 2: Create a variable named straincentroids which contains the centroids of the 11 different strains. Step 3: Add the 11 centroids of the different strains to the plot. Make them large black dots. Step 4: Create a variable named strainchangemagnitudes which contains the magnitude of vector which points from the centroid of strain i to the centroid of strain i + 1. These indicate how much antigenic change occured between strains. 4.2 Questions: Question 1: Between which two strains did the largest antigenic change occur? Question 2: What was the total antigenic change between 1968 and 2003 (i.e. between strains 1 and 11)? Question 3: Compute and record the vector which points from the centroid of strain 3 to strain 4 and the vector which points from strain 4 to strain 5 to at least 6 demical places. Compute the sum of these two vectors and explain this sum of vectors in terms of the antigenic change between the centroids of strains. Question 3: Print the scatter plot antigenic data and strain centroids and draw and label (by hand) the vectors described in question 3. Bonus: The triangle inequality for vectors states that a + b a + b. Verify this claim by computing a+b and a + b if a is the difference between the centroid of strain 11 and the centroid of strain 10 and if b is the difference between the centroid of strain 10 and the centroid of strain 9. Explain what the triangle inequality means in terms of antigenic change. 7