Chapter2

Chapter2 - Chapter 2 Vector Spaces Keith E Emmert Euclidean...

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Unformatted text preview: Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Chapter 2: Vector Spaces Keith E. Emmert Tarleton State University January 13, 2011 Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Overview Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Common Notation Definition 1 I N = { 1 , 2 , 3 ,... } is the set of natural numbers . I W = { , 1 , 2 , 3 ,... } is the set of whole numbers . I Z = {- 3 ,- 2 ,- 1 , , 1 , 2 , 3 ,... } is the set of integers . I Q = n a b | a , b ∈ Z , b 6 = o is the set of rational numbers . I R is the set of real numbers . I Let i = √- 1. Then C = { a + bi | a , b ∈ R } is the set of complex numbers . I The set of polynomials whose degree is less than or equal to n is P n = { a + a 1 t + a 2 t 2 + ··· + a n t n | a , a 1 ,..., a n ∈ R } . I M m × n is the set of all (real valued) matrices of size m × n ( m rows and n columns). Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Vector Space Defined Definition 2 Let set F be a set of elements, scalars , with operations , (addition & multiplication). A vector space is a set V of elements, vectors , with vector addition , ⊕ , and scalar multiplication , , satisfying the ten axioms: Additive Properties 1. Closure for Addition If u , v ∈ V , then u ⊕ v ∈ V . 2. Commutativity of Addition If u , v ∈ V , then u ⊕ v = v ⊕ u . 3. Associativity of Addition If u , v , w ∈ V , then ( u ⊕ v ) ⊕ w = u ⊕ ( v ⊕ w ) . 4. Existence of a Zero Vector There exists ∈ V such that for all u ∈ V , u ⊕ = u . 5. Existence of Additive Inverses For all u ∈ V , there exists at least one element ˜ u ∈ V such that ˜ u ⊕ u = . Fact: ˜ u is unique! Let ˜ u =- u . Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Vector Space Definition Continued Definition 2 (Continuation of Vector Space Definition) Product Properties 1. Closure for Scalar–Vector Product If α ∈ F and u ∈ V , then α u ∈ V . 2. Distributive Law: Scalar Times Vectors For any α ∈ F and u , v ∈ V , we have α ( u ⊕ v ) = ( α u ) ⊕ ( α v ) . 3. Distributive Law: Scalar Sum Times Vector For any α,β ∈ F and u ∈ V , we have ( α β ) u = ( α u ) ⊕ ( β u ) ....
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Chapter2 - Chapter 2 Vector Spaces Keith E Emmert Euclidean...

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