214 CHAPTER 4 Finite Dimensional Vector Sp Definition A vector space is a set V

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214 CHAPTER 4 Finite Dimensional Vector Sp Definition A vector space is a set V together with an addition Ca rule for adding two elements of V to and another operation called scalar multiplication Ca rule number times an element of V to obtain a second eleme element of v on which the for Examp multi For every u, v, and w in Y, and for every a and b in R (A) u+vev (B) au e V Exampl Let V Sectio prope as we following ten properties hold: Closure Property of Addition Closure Property of Scalar Multiplication Commutative Law of Associative Law of Addition Sir additi the c u+(v + w) = (u + v) + w There is an element 0 of V so that for every y in V we have (2) (3) Existence of Ident for Addition ) There is an element - u in V such Existence of Additiv tion con that u + (-u) = 0 = (-u) + u. (5) a(u + v) = (au) + (av) (6) (a + b)u = (au) + (bu) (7) (ab)u=a(bu) Distributive Laws for Scalar Associativity of Scalar Identity Property for Scalar Multiplication over Addition Multiplication Multiplication ple But the ad on (8) 1u = u The elements of a vector space V are called vectors. and scalar multiplication always produce an element of the vector space as a result. The two closure properties require that both the operations of vector addition Ex TI All sums indicated by "+" in properties (1) through (5) are vector sums. In property (6), the “+" on the left side of the equation represents addition of real numbers, the +on the right side stands for the sum of two vectors. In property , the left side of the equation contains one product of real numbers, ab, and one instance of scalar multiplication, (ab) times u. The right side of property (7) involves two multiplications- first, b times u, and then, a times the vector (bu). Usually w tellfrom the context which type of operation is involved. In any vector space, the additive identity element in property (3) is unique, the additive inverse (property ) of each vector is unique (see the proof of Part of Theorem 4.1 and Exercise 11). can

Explanation / Answer

(a) Every vector space can be a Canadian vector space, because the condition (4) there is simply about scalar multiplication, which every other non-Canadian vector space satisfies.

(b) The condition (4) given on page 214 is essential to satisfy the closure property of vector addition. Since the Canadian vector space does not have this property, it cannot be a vector space. For example if vector w is an element of Canadian vector space W, it will not have the vector -w . This means the Canadian vector space does not have 0 vector as an element. In other words, in the Canadian vector space w +(-w) will not be a 0 vector

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