CH2p1.pdf - Math 4242 Applied Linear Algebra Vector Spaces...

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Math 4242 - Applied Linear Algebra Vector Spaces, Bases, and Span these notes follow closely Olver and Shakiban 2.1-2.3 Natalie E. Sheils last update: September 23, 2015 1 Vector Spaces A vector space is a set V equipped with two operations: 1. Addition: adding any two vectors ~v, ~w V gives another vector ~v + ~w V . 2. Scalar Multiplication: multiplying a vector ~v by a scalar c R produces another vector c~v V . subject to the following axioms for all ~u,~v, ~w V and all scalars c, d R : (a) Commutativity of addition: ~v + ~w = ~w + ~v. (b) Associative of addition: ~u + ( ~v + ~w ) = ( ~u + ~v ) + ~w. (c) Additive identity: There is a zero element ~ 0 V satisfying ~v + ~ 0 = ~v = ~ 0 + ~v. (d) Additive inverse: For each ~v V there is an element - ~v V such that ( - ~v )+ ~v = ~ 0 = ~v +( - ~v ) . (e) Distributivity: ( c + d ) ~v = c~v + d~v and c ( ~v + ~w ) = c~v + c~w. (f) Associativity of scalar multiplication: c ( d~v ) = ( cd ) ~v. (g) Unit for Scalar Multiplication: the scalar 1 R satisfies 1 ~v = ~v . This is an abstraction of what we are used to working in, R n , the set of all n -dimensional vectors. To this point, the set of all n × m matrices is a vector space and we showed this on the first day of class. Example 1. Consider the space P ( n ) = { p ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 1 x + a 0 } , the set of polynomials of degree less than or equal to n . Clearly the sum of two polynomials p ( x ) and q ( x ) of degree less than or equal to n is another polynomial of degree less than or equal to n . Scalar multiplication works just the way we expect it to. 1
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Example 2. Let I R be an interval, say [ a, b ] . Pick F = { all functions f : I R } . Let’s verify that F is indeed a vector space. Let f, g, h ∈ F and c, d R . We need addition: f ( x ) + g ( x ) is still a function which takes as input values on [ a, b ] and outputs real numbers. Thus, f ( x )+ g ( x ) F . Next we need scalar multiplication. In fact, cf ( x ) takes as input values on [ a, b ] and outputs real numbers. Thus, cf ( x ) ∈ F . Now lets check the remaining seven properties: (a) Commutativity of addition: f ( x ) + g ( x ) = g ( x ) + f ( x ) . (b) Associative of addition: f ( x ) + ( g ( x ) + h ( x )) = ( f ( x ) + g ( x )) + h ( x ) . (c) Additive identity: The zero element is the “zero function” f ( x ) + 0 = f ( x ) . (d) Additive inverse: f ( x ) + ( - f ( x )) = 0 . Since - f ( x ) ∈ F as well. (e) Distributivity: ( c + d ) f ( x ) = cf ( x ) + df ( x ) and c ( f ( x ) + g ( x )) = cf ( x ) + cg ( x ) .
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