Vectors, Matrices and Geometry

CHAPTER FIVE: HIGHER DIMENSIONAL VECTOR SPACES

CHAPTER FIVE HIGHER DIMENSIONAL VECTOR SPACES In the first two chapters, we have learnt the language and techniques of linear algebra of vectors in R2 and R3. Instead of moving up one dimension from R3 to R4, we shall study the general n-dimensional vector space Rn for any positive integer n. However the nature of the present course only allows us a restricted scope of study. We shall therefore concentrate on the notions of linear independence and of subspace of Rn. In order to study the notion of dimension properly, we find it necessary to introduce matrices and elementary transformations on matrices. 5.1 The vector space Rn Let n be any positive integer. If n =2, then an n-dimensional vector is an order pair [Xl, X2] of real numbers. If n = 3, then an n-dimensional vector is an order triple [Xl, X2, X3] of real numbers. For an arbitrary positive integer n, an n-dimensional vector is an ordered n-tuple x =[Xl, X2,' .. ,xn ] of real numbers Xi. The real number Xi, i = 1,2"" ,n is called the i-th component of the vector x = [XI,X2,'" ,xn]' Thus two n-dimensional vectors x = [XI,X2,'" ,xn] and Y = [YI, Y2, .. , ,Yn] are equal if and only if they have identical components: x =y if and only if Xi =Yi for i =1,2" .. ,n. The set of all n-dimensional vectors shall be denoted by Rn: We take note that here we continue to follow the convention of the earlier chapters by using lower case bold-faced types to denote vectors and brackets to enclose the components of a vector. For example [1, -1,0,1] is a 4-dimensional vector and [1, -1,0,1] E R4 while [1, -1, 0, 1,0,0] is a 6-dimensional vector and [1, -1,0, 1,0,0] E RB. 207 Vectors, Matrices and Geometry In Rn the n vectors el = [1,0,··· ,0], e2 = [0,1,0,··· ,0],···, en = [0,··· ,0,1] are called the unit coordinate vectors of Rn. We note that the i-th unit coordinate vector ei has the i-th component equal to 1 and all other components equal to zero. For example the 3-rd unit vector of R4 is [0,0,1,0] while the 2-nd unit vector of R6 is the 6-tuple [0,1,0,0,0,0]. To express the components of the unit coordinate vectors more conveniently we can make use of the Kronecker symbols bij which is defined by { o if i#j bij = 1 if i =j for all i = 1,2, ... ,n and j = 1,2, ... ,n. For example 823 = 816 = °and b11 = 833 = 1. Now the i-th unit coordinate vector of Rn is simply Next we specify the sum of two vectors x = [Xl, X2, ... ,xn ] and Y = [Yl, Y2,· .. ,Ynl of Rn to be the vector x + Y = [Xl + Yl, X2 + Y2,··· ,Xn + Ynl . With respect to the addition of vectors, the zero vector 0 = [0,0, ... ,OJ, whose n components are all zero, plays the role of the additive identity in the sense that x + 0 = 0 +x = x for all vectors of Rn. Finally the scalar multiple rx of a vector x of Rn by a real number r is the vector If r = 1, then Ix = x. If r = -1, we denote the scalar multiple (-l)x = [-Xl, -X2,··· , -xnl by -x. The vector -x is then the additive inverse of x in the sense that x + (-x) = (-x) +x = o. Now the set Rn of all n-dimensional vectors together with the addition and the scalar multiplication will be called the n-dimensional vector space Rn. For n = 1, RI is essentially the same as the system R of real numbers. For n = 2 and n = 3, R2 and R3 are just the vector space of plane vectors and the vector space of space vectors respectively. Furthermore, like R2 and R3, many properties of the 208 Higher Dimensional Vector Spaces n-dimensional vector space Rn can be directly derived from the fundamental properties contained in the follow theorem. 5.1.1 THEOREM Let a, b, c be vectors of the vector space Rn, and r, s be scalars. Then the following statements hold. (1) a + b =b + a. (2) (a +b) + c = a + (b + c). (3) There exists a unique...