Lecture 3: Vectors. In Song Kim. September 1, 2011

Transcription

1 Lecture 3: Vectors In Song Kim September 1, Solving Equations Up until this point we have been looking at different types of functions, often times graphing them. Each point on a graph is a solution to that equation. Now we will look at cases where multiple equations are used to describe a relationship. These are known as systems of equations. 1.1 Systems of Linear Equations In general a system of n linear equations with n variables is solvable. There are 2 general methods of solution: 1. Substitution Example: x + 2y = 3 x y = 4 Solve for x in terms of y in the second equation: x = 4 + y Substitute this into the first equation, and solve for y Now calculate x = 4 + ( 1/3) = 11/3 2. Cancelation (4 + y) + 2y = y = 3 y = 1/3 x + 2y = 3 x y = 4 Multiply both sides of one of the equations by a number such that one of the variables cancels out when you sum them up. Please do not distribute without permission. Ph.D. candidate, Department of Politics, Princeton University, Princeton NJ insong@princeton.edu 1

2 In this case, multiply the second equation by 2: x + 2y = 3 2x 2y = 8 3x = 11 x = 11/3 To get y, substitute the value for x into one of the equations: 11/3 + 2y = 3 y = 1/3 3. Note: This trick can also be used in some non-linear systems. For example try it on this one: You should get y = 3, x = ± 3 x 2 + y = 6 2x 2 y = Properties of systems of linear equations Systems of linear equations fall into one of three camps A single solution (the two lines intersect at one point) No solution (the two lines are exactly parallel but are shifted by some constant) An infinite number of solutions If you are working with a system of linear equations and you get exactly two solutions, then something is wrong Practice Problems galore 3x + 2y = 4 x 2y = 8 x = 3 2, y = 1 4 x 2y = 3 2x + 4y = 1 no solution x 2 x y = 1 x + y = 1 x =, 2 2

3 x + y = 4 x2 + y = 3 no real solution 1.4 Graphical View of the Solutions Consider the following three systems equations. x+y+z =3 (1) x + 2y 3z = 2 (2) x+y+z = (3) Figure 1: Left-hand panel shows that there exists infinite number of solutions that satisfy both Equation (1) and Equation (2). However, the right-hand panel shows that there exists no solution that satisfies all the equations from (1) to (3). Notice that the plane represented by Equation (1) and Equation (3) are parallel with each other. 2 Vectors, Victors, and Matrices Up until now we have been using the symbols x and y to represent our variables. We are now going to begin shifting our notation, and using indices to label our variables. Here is a previous example, but we now use the x variable with different integers to index and indicate that they are different variables. You would as you normally would. x1 + 2x2 = 3 x1 x2 = 4 Now instead of having the x, y coordinate system, we have the x1, x2 coordinate system. In the next section we will learn about how to represent these as vectors. Lets say there are two key pieces of information about an individual that we need to know: whether they are Republican (1) or Democrat (-1) and whether they voted in the last election (1) or 3

4 did not vote (-1). Hence for every individual we can define a vector that contains this information. In this case the vector will be in R 2. The form of the vector is (Party,Vote). We could, of course, include many other characteristics of an individual and get a vector in R n (Party,Vote,Gender,Age...). The way we will typically think of vectors is that they store information about observations. Physicists think of vectors in physical terms, which will be helpful for us later on. Example 1 (1, 2), (, 1), (, ), (3, 1, 2) 2.1 Definition of Vector Vector: A vector in n-space is an ordered list of n numbers. These numbers can be represented as either a row vector or a column vector: v 1 v = ( ) v 2 v 1 v 2... v n, v =. v n We can also think of a vector as defining a point in n-dimensional space, usually R n ; each element of the vector defines the coordinate of the point in a particular direction. 2.2 Properties of Vectors Vector Addition: Vector addition is defined for two vectors u and v iff ( if and only if ) they have the same number of elements and are either both row or both column vectors: u + v = ( u 1 + v 1 u 2 + v 2 u k + v n ) Try these examples (1, 3, 4) + (3, 4, 2) + (1, 1, 1) (6, 2, 2) + (3, 1, 1) + (1, 1) Geographic representation on board ( displacement arrows ): (1, 2) + (, 1) = 1, 1 a, b, a + b Scalar Multiplication: The product of a scalar c and vector v is: cv = ( cv 1 cv 2... cv n ) All we are doing here is multiplying every vector by a constant Imagine having a set of data on income in pesos, but you want income in US dollars. You could multiply by the exchange rate r v = (3, 2, 1) find rv if r = 1 1 4

5 Vector Inner Product: The inner product (also called the dot product or scalar product) of two vectors u and v is again defined iff they have the same number of elements. n u v = u 1 v 1 + u 2 v u n v n = u i v i If u v =, the two vectors are orthogonal (or perpendicular). following vectors in R 2 and calculate their inner product. i=1 For example, draw the (1, ), (, 1) ( 3, 2), (2, 3) Vector Norm: The norm of a vector is a measure of its length. We will often use the Euclidean norm (which corresponds to our usual conception of distance in three-dimensional space): v = v v = v 1 v 1 + v 2 v v n v n Think about the Pythagorean theorem in R 2... Connection to the triangle inequality using a, b, a + b Class exercizes 3 Functions of Vectors Sometimes it is possible to write one vector as function of other vectors. E.g., the amount someone contributes to McCain s campaign is a function of their party affiliation, gender, income, and past vote. Here we take the vector (P, G, I, V ) multiply each term by its own special weight, and then add them up. C = β 1 P + β 2 G + β 3 I + β 4 V In particular, we ve taken a linear combination of our independent variables. Now imagine that we had data on more than one person, such that for every individual we had the data (C, P, G, I, V ). We could then write out each element as a vector across individuals: C P G I V C = β 1 P + β 2 G + β 3 I + β 4 V 2 P = 7 6

6 3.1 Definitions Linear combinations: The vector u is a linear combination of the vectors v 1, v 2,, v k if u = c 1 v 1 + c 2 v c k v k Linear independence: A set of vectors v 1, v 2,, v k is linearly independent if the only solution to the equation c 1 v 1 + c 2 v c k v k = is c 1 = c 2 = = c k =. If another solution exists, the set of vectors is linearly dependent. A set S of vectors is linearly dependent iff at least one of the vectors in S can be written as a linear combination of the other vectors in S. Linear independence is only defined for sets of vectors with the same number of elements; any linearly independent set of vectors in n-space contains at most n vectors. Example v 1 = 2, v 2 =, v 3 = We want to check if there are values of c 1, c 2, c 3 such that c c 2 + c 3 8 = answ:1,-2,1 and hence is linearly dependent. Exercises: Are the following sets of vectors linearly independent? v 1 =, v 2 =, v 3 = v 1 = 2, v 2 = 2, v 3 = v 1 =, v 2 =, v 3 = 1 1 With our understanding of what vectors are, how to manipulate them, and some simple properties, we will now collect vectors together in order to form matrices. 6

7 4 Linear Equations We have seen all of the before, but now we are ready to express them with some more notation. Linear Equation: a 1 x 1 + a 2 x a n x n = b a i are parameters or coefficients. x i are variables or unknowns. Linear because only one variable per term and degree is at most R 2 : line x 2 = b a 2 a 1 a 2 x 1 2. R 3 : plane x 3 = b a 3 a 1 a 3 x 1 a 2 a 3 x 2 3. R n : hyperplane Systems of Linear Equations Often interested in solving linear systems More generally, we might have a system of m equations (one ser of values for every individual) with n unknowns a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2... a m1 x 1 + a m2 x a mn x n = b m A solution to a linear system of m equations in n unknowns is a set of n numbers x 1, x 2,, x n that satisfy each of the m equations. 1. R 2 : intersection of the lines. 2. R 3 : intersection of the planes. 3. R n : intersection of the hyperplanes. 6 Method of Solving Systems (continued) We previously went over how to solve a system by substitution. For a refresher, use substitution to solve: Other ways are possible... x + 2y + 3z = 6 2x 3y + 2z = 14 3x + y z = 2 7