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**Unformatted text preview: **Chapter 3. Vector Spaces 3.1. Examples and Basic Properties Vector spaces are the basic objects of study in Linear Algebra. A vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms. The most familiar vector spaces are R 2 , R 3 or more generally R n . Vectors in these spaces are often represented as geometric vectors which are quantities with a magnitude and a direction. These vectors may be added together (vector addition) or multiplied by real numbers: Here is the formal definition of a vector space: Definition 3.1 A vector space is a set V together with two binary operations, namely the vector addition and scalar multiplication, such that (a) for any u , v ∈ V , we have u + v ∈ V , and (b) for any α ∈ R and u ∈ V , we have α u ∈ V , and the following axioms are satisfied. (1) u + v = v + u for any u , v ∈ V , (2) ( u + v ) + w = u + ( v + w ) for any u , v , w ∈ V , (3) There exists an element ∈ V , called the zero vector, such that u + = u for all u ∈ V , (4) For all u ∈ V , there exists an element v ∈ V , called the additive inverse of u , such that u + v = , (5) For all α ∈ R and u , v ∈ V , we have α ( u + v ) = α u + α v , (6) For all α, β ∈ R and u ∈ V , we have ( α + β ) u = α u + β u , (7) For all α, β ∈ R and u ∈ V , ( αβ ) u = α ( β u ), (8) For all u ∈ V , we have 1 · u = u . Elements in V are generally referred to as vectors . Conditions (a) and (b) are called the closure axioms . Let us look at an example of a set which is not closed under both addition and scalar multiplication: Example 3.2 Consider the set V = { (1 , y ) T : y ∈ R } with standard addition and scalar multiplication in R 2 . Then (1 , y 1 ) T +(1 , y 2 ) T = (2 , y 1 + y 2 ) T is not an element of V . Moreover, if we take α = 3 , then 3 · (1 , y ) T = (3 , 3 y ) T is not an element of V again. Therefore (a) and (b) fail to hold on V and hence the set V with the operations of addition and scalar multiplication is NOT a vector space. 2 Here is another example of a set which is not a vector space: Example 3.3 Let V be the set R 2 with the standard addition. Define scalar multiplication on V by α ⊗ ( x, y ) T = ( αx, y ) T . This is an unusual way of defining a scalar multiplication and so we use the symbol ⊗ to distinguish the ordinary scalar multiplication. It is not hard to see that V is closed under addition and the scalar multiplication ⊗ . However, if we look at axiom (6) from Definition 3.1, then ( α + β ) ⊗ ( x, y ) T = (( α + β ) x, y ) T = ( αx + βx, y ) T , whereas α ⊗ ( x, y ) T + β ⊗ ( x, y ) T = ( αx, y ) T + ( βx, y ) T = ( αx + βx, 2 y ) T ....

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