Notes for CIT Mathematics 1c 1 n-Dimensional Euclidean Space and Matrices Version: April, 2008 Definition of n space. As was learned in Math 1b, a point in Euclidean three space can be thought of in any of three ways: (i) as the set of triples (x; y; z) where x; y, and z are real numbers; (ii) as the set of points in space; (iii) as the set of directed line segments in space, based at the origin. The first of these points of view is easily extended from 3 to any number n of dimensions. We define R , where n is a positive integer (possibly greater than 3), as the set of all orderedpn-tuples (x1; x2; : : : ; xn), where the xi are 4 real numbers. For instance, (1; 5; 2; 4) 2 R . n The set R is known as Euclidean n-space, and we may think of its elements a = (a1; a2; : : : ; an) as vectors or n-vectors. By setting n = 1; 2, or 3, we recover the line, the plane, and three-dimensional space respectively. Addition and Scalar Multiplication. We begin our study of Euclidean n-space by introducing algebraic operations analogous to those learned for 2 3 R and R . Addition and scalar multiplication are defined as follows: (i) (a1; a2; : : : ; an) + (b1; b2; : : : ; bn) = (a1 + b1; a2 + b2; : : : ; an + bn); and (ii) for any real number α, α(a1; a2; : : : ; an) = (αa1; αa2; : : : ; αan): 2 3 The geometric significance of these operations for R and R is discussed in \Calculus", Chapter 13. 2 Informal Discussion. What is the point of going from R; R 3 n and R , which seem comfortable and \real world", to R ? Our world is, after all, three-dimensional, not n-dimensional! First, it is true that the bulk of multivariable calculus is about 2 3 n R and R . However, many of the ideas work in R with little extra effort, so why not do it? Second, the ideas really are useful! For instance, if you are studying a chemical reaction involving 5 chemicals, you will probably want to store and manipulate their 5 concentrations as a 5-tuple; that is, an element of R . The laws governing chemical reaction rates also demand we do calculus in this 5-dimensional space. Notes for CIT Mathematics 1c 2 n n The Standard Basis of R . The n vectors in R defined by e1 = (1; 0; 0;:::; 0); e2 = (0; 1; 0;:::; 0);:::; en = (0; 0;:::; 0; 1) n are called the standard basis vectors of R , and they generalize the three 3 mutually orthogonal unit vectors i; j; k of R . Any vector a = (a1; a2; : : : ; an) can be written in terms of the ei's as a = a1e1 + a2e2 + ::: + anen: Dot Product and Length. Recall that for two vectors a = (a1; a2; a3) 3 and b = (b1; b2; b3) in R , we defined the dot product or inner product a · b to be the real number a · b = a1b1 + a2b2 + a3b3: n This definition easily extends to R ; specifically, for a = (a1; a2; : : : ; an), and b = (b1; b2; : : : ; bn), we define a · b = a1b1 + a2b2 + ::: + anbn: We also define the length or norm of a vector a by the formula p q 2 2 2 length of a = kak = a · a = a1 + a2 + ::: + an: The algebraic properties 1{5 (page 669 in \Calculus", Chapter 13) for n the dot product in three space are still valid for the dot product in R . Each can be proven by a simple computation. Vector Spaces and Inner Product Spaces. The notion of a vector space focusses on having a set of objects called vectors that one can add and multiply by scalars, where these operations obey the familiar rules of n vector addition. Thus, we refer to R as an example of a vector space (also called a linear space). For example, the space of all continuous functions f defined on the interval [0; 1] is a vector space. The student is familiar with how to add functions and how to multipy them by scalars and they obey similar algebraic properties as the addition of vectors (such as, for example, α(f + g) = αf + αg). This same space of functions also provides an example of an inner prod- uct space, that is, a vector space in which one has a dot product that sat- isfies the properties 1{5 (page 669 in \Calculus", Chapter 13. Namely, we define the inner product of f and g to be Z 1 f · g = f(x)g(x) dx: 0 Notes for CIT Mathematics 1c 3 Notice that this is analogous to the definition of the inner product of two n vectors in R , with the sum replaced by an integral. The Cauchy-Schwarz Inequality. Using the algebraic properties of the dot product, we will prove algebraically the following inequality, which is 3 normally proved by a geometric argument in R . Cauchy-Schwarz Inequality n For vectors a and b in R , we have ja · bj ≤ kak kbk: If either a or b is zero, the inequality reduces to 0 ≤ 0, so we can assume that both are non-zero. Then 2 a b 2a · b 0 ≤ − = 1 − + 1; kak kbk kak kbk that is, 2a · b 0 ≤ − : kak kbk Rearranging, we get a · b ≤ kak kbk: The same argument using a plus sign instead of minus gives −a · b ≤ kak kbk: The two together give the stated inequality, since ja · bj = ±a · b de- pending on the sign of a · b. If the vectors a and b in the Cauchy-Schwarz inequality are both nonzero, a·b the inequality shows that the quotient kak kbk lies between −1 and 1. Any number in the interval [−1; 1] is the cosine of a unique angle θ between 0 and π; by analogy with the 2 and 3-dimensional cases, we call a · b θ = cos−1 kak kbk the angle between the vectors a and b. Informal Discussion. Notice that the reasoning above is oppo- 3 site to the geometric reasoning normally done in R . That is be- n cause, in R , we have no geometry to begin with, so we start with algebraic definitions and define the geometric notions in terms of them. Notes for CIT Mathematics 1c 4 The Triangle Inequality. We can derive the triangle inequality from the Cauchy-Schwarz inequality: a · b ≤ ja · bj ≤ kak kbk, so that ka + bk2 = kak2 + 2a · b + kbk2 ≤ kak2 + 2kakkbk + kbk2: Hence, we get ka+bk2 ≤ (kak+kbk)2; taking square roots gives the following result. Triangle Inequality n Let vectors a and b be vectors in R . Then ka + bk ≤ kak + kbk: If the Cauchy-Schwarz and triangle inequalities are written in terms of components, they look like this: ! 1 ! 1 n n 2 n 2 X X 2 X 2 aibi ≤ ai bi ; i=1 i=1 i=1 and 1 1 1 n ! 2 n ! 2 n ! 2 X 2 X 2 X 2 (ai + bi) ≤ ai + bi : i=1 i=1 i=1 The fact that the proof of the Cauchy-Schwarz inequality depends only on the properties of vectors and of the inner product means that the proof n works for any inner product space and not just for R . For example, for the space of continuous functions on [0; 1], the Cauchy-Schwarz inequality reads s s Z 1 Z 1 Z 1 f(x)g(x) dx ≤ [f(x)]2 dx [g(x)]2 dx 0 0 0 and the triangle inequality reads s s s Z 1 Z 1 Z 1 [f(x) + g(x)]2 dx ≤ [f(x)]2 dx + [g(x)]2 dx: 0 0 0 Notes for CIT Mathematics 1c 5 Example 1. Verify the Cauchy-Schwarz and triangle inequalities for a = (1; 2; 0; −1) and b = (−1; 1; 1; 0). Solution. p p kak = 12 + 22 + 02 + (−1)2 = 6 p p kbk = (−1)2 + 12 + 12 + 02 = 3 a · b = 1(−1) + 2 · 1 + 0 · 1 + (−1)0 = 1 a + b = (0; 3; 1; −1) p p ka + bk = 02 + 32 + 12 + (−1)2 = 11: p p We compute a · b = 1 and kak kbk = 6 3 ≈ 4:24, which verifies the Cauchy-Schwarz inequality. Similarly, we can check the triangle inequality: p ka + bk = 11 ≈ 3:32; while p p kak + kbk = 2:45 + 1:73 ≈ 6 + 3 ≈ 4:18; which is larger. 3 Distance. By analogy with R , we can define the notion of distance in n n R ; namely, if a and b are points in R , the distance between a and b is defined to be ka − bk, or the length of the vector a − b. There is no cross n 3 product defined on R except for n = 3, because it is only in R that there is a unique direction perpendicular to two (linearly independent) vectors. It is only the dot product that has been generalized. Informal Discussion. By focussing on different ways of thinking about the cross product, it turns out that there is a generalization of n the cross product to R . This is part of a calculus one learns in more advanced courses that was discovered by Cartan around 1920. This calculus is closely related to how one generalizes the main integral n theorems of Green, Gauss and Stokes' to R . Matrices. Generalizing 2 × 2 and 3 × 3 matrices, we can consider m × n matrices, that is, arrays of mn numbers: 2 3 a11 a12 : : : a1n 6 a21 a22 : : : a2n 7 A = 6 7 : 6 .