l9 - Math 20F Linear Algebra Lecture 9: 4.1 Vector Spaces....

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Math 20F Linear Algebra Lecture 9: 4.1 Vector Spaces. Recall that if x , y R n are vectors in Euclidean space we defined the addition x + y R n and scalar multiplication λ x R n . In dimensions 2 and 3 we can define these notions geometrically The addition and scalar multiplication satisfy certain properties listed below, but these properties show up in many different contexts so rather than studying each situation individually we will study them all at once. A set V with two operations, addition and multiplication by scalars, defined on it is called a vector space if the following properties hold for any u , v , w V α,β R : 1. If u , v V then u + v V . (closure under addition) 2. u + v = v + u (commutative) 3. ( u + v ) + w ) = u + ( v + w ) (associative) 4. There is an element 0 V such that u + 0 = u all u V (additive unit) 5. For each u V there is - u V such that u + ( - u ) = 0 (additive inverse) 6. If u V and α is a scalar then α u V . (closure under scalar multiplication)
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l9 - Math 20F Linear Algebra Lecture 9: 4.1 Vector Spaces....

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