Vector_Spaces - EECS 221 A Vector Spaces A Notation B...

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Unformatted text preview: EECS 221 A Vector Spaces A Notation B Algebraic Aspects C Normed Vector Spaces D Inner Product Spaces E The Projection Theorem 1 A. Notation ∃ there exists ∃ ! there exists a unique ∀ for all R field of real numbers C field of complex numbers Q field of rational numbers R n , C n vector space of n tuples of real or complex numbers R m × n C m × n vector space of m by n matrices of real or complex numbers R ( s ) field of rational functions in s with real coefficients R [ s ] ring of polynomials in s with real coefficients A the transpose of the matrix A A * the adjoint of the operator A , or the complex-conjugate-transpose of the matrix A A function f : X → Y is called an injection if f ( x 1 ) = f ( x 2 ) ⇐⇒ x 1 = x 2 The function f is called a surjection if ∀ y ∈ Y, ∃ x ∈ X such that f ( x ) = y The function f is called a bijection if it is both surjective and injective. 2 B. Algebraic Aspects 1 Definition A field F is a set of elements called scalars together with two binary op- erations, addition (+) and multiplication ( · ) such that for all α,β,γ ∈ F the following properties hold: (a) Closure. α · β ∈ F , α + β ∈ F (b) Commutativity. α · β = β · α, α + β = β + α (c) Associativity. α + ( β + γ ) = ( α + β ) + γ, α · ( β · γ ) = ( α · β ) · γ (d) Distribution. α · ( β + γ ) = ( α · β ) + ( α · γ ) (e) Identity. There exists an additive identity ∈ F and a multiplicative identity 1 ∈ F such that α + 0 = α, α · 1 = α (f) Inverses. For all α ∈ F there exists an additive inverse- α ∈ F such that α + (- α ) = 0. For all α ∈ F , α 6 = 0 and a multiplicative inverse α- 1 ∈ F such that α · α- 1 = 1 2 Examples The following are examples of fields: ƒ R = the set of real numbers ƒ C = the set of complex numbers ƒ Q = the set of rational numbers ƒ R ( s ) = the set of rational functions in s with real coefficients These are not fields: ƒ R [ s ] = the set of polynomials in s with real coefficients. Why? ƒ R 2 × 2 the set of real 2 × 2 matrices. Why? 3 Definition A vector space ( V , F ) is a set of vectors V together with a field F and two operations vector-vector addition (+) and vector-scalar multiplication ( ◦ ) such that for all α,β ∈ F and all v 1 ,v 2 ,v 3 ∈ V the following properties hold: (a) Closure. v 1 + v 2 ∈ V , α ◦ v 1 ∈ V (b) Commutativity. v 1 + v 2 = v 2 + v 1 (c) Associativity. ( v 1 + v 2 ) + v 3 = v 1 + ( v 2 + v 3 ) (d) Distribution. α ◦ ( β ◦ v 1 ) = ( α · β ) ◦ v 1 , α ◦ ( v 1 + v 2 ) = α ◦ v 1 + α ◦ v 2 (e) Additive Identity. There exists a vector 0 ∈ V such that v + 0 = v for all v ∈ V (f) Additive Inverse. For all v ∈ V , there exists a (- v ) ∈ V such that v + (- v ) = 0 3 We shall henceforth suppress the cumbersome notation · , ◦ as the appropriate action will be clear from context. Also, we shall often refer to a vector space V without explicit reference to the base field F (which will exclusively be...
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This note was uploaded on 09/29/2009 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Vector_Spaces - EECS 221 A Vector Spaces A Notation B...

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