la_m10_handouts

la_m10_handouts - MAC 2103 Module 10 lnner Product Spaces I...

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1 1 MAC 2103 Module 10 lnner Product Spaces I 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Define and find the inner product, norm and distance in a given inner product space. 2. Find the cosine of the angle between two vectors and determine if the vectors are orthogonal in an inner product space. 3. Find the orthogonal complement of a subspace of an inner product space. 4. Find a basis for the orthogonal complement of a subspace of ℜⁿ spanned by a set of row vectors. 5. Identify the four fundamental matrix spaces of an m x n matrix A, and know the column rank of A, the row rank of A, and the rank of A. 6. Know the equivalent statements of an n x n matrix A. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
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2 3 Rev.09 General Vector Spaces II http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Inner Product Spaces, Inner Product Spaces, Inner Products, Inner Products, Norm, Distance, Fundamental Matrix Spaces, Norm, Distance, Fundamental Matrix Spaces, Rank, and Orthogonality Orthogonality The major topics in this module: 4 Rev.F09 Definition of an Inner Product and Orthogonal Vectors http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. A. Inner Product: Let be any function from into that satisfies the following conditions for any u , v, z in V: and iff Then defines an inner product for all u , v in V. u and v in V are Orthogonal Vectors iff !" , "# : V $ V % ! V ! V ! c ) ! k ! u , ! v " = k ! ! u , ! v " , a ) ! ! u , ! v " = ! ! v , ! u " , b ) ! ! u + ! v , ! z " = ! ! u , ! z " + ! ! v , ! z " , d ) ! ! v , ! v " # 0, ! ! v , ! v " = 0 ! v = 0. ! ! u , ! v " ! ! u , ! v " = 0.
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3 5 Rev.F09 Definition of a Norm of a Vector http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. B. Norm: Let be any function from into that satisfies the following conditions for any u , v in V: Then defines a norm for all u in V. The special norm that is induced by an inner product is ! : V " ! + = [0, # ) V ! + ! u a ) ! u ! 0, and ! u = 0 iff ! u = ! 0, b ) s ! u = s ! u , and c ) ! u + ! v " ! u + ! v Triangle inequality . ! u = ! ! u , ! u " . 6 Rev.F09 Definition of the Distance Between Two Vectors http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. C. Distance: Let be any function from into that satisfies the following conditions for any u , v, w in V: Then defines the distance for all u , v in V. The special distance that is induced by a norm is d ( ! , ! ) : V " V # ! + = [0, $ ) V ! V ! + d ( ! u , ! v ) d ( ! u , ! v ) = ! u ! ! v = ! u ! ! v , ! u ! ! v # . a ) d ( ! u , ! v ) ! 0, and d ( ! u , ! v ) = 0 iff ! u = ! v , b ) d ( ! u , ! v ) = d ( ! v , ! u ), and c ) d ( ! u , ! v ) " d ( ! u , ! w ) + d ( ! w , ! v ) Triangle inequality .
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4 7 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors in ℜⁿ ? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. For u and v in ℜⁿ , we define and Note: These are our usual norm and distance in ℜⁿ . ℜⁿ is an inner product space with the dot product as its inner product. ! u = ! ! u , ! u " 1 2 = ! u # ! u ) = 1 2 + u 2 2 + ... + u n 2 , d ( ! u , ! v ) = ! u ! ! v = " ! u ! ! v , ! u ! ! v # 1 2 = ( ! u ! ! v ) $ ( ! u ! ! v ) = u 1 ! v 1 ) 2 + ( u 2 ! v 2 ) 2 + ... + ( u n ! v n ) 2 .
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This note was uploaded on 10/18/2011 for the course MAS 2103 taught by Professor Shaw during the Summer '11 term at Valencia.

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la_m10_handouts - MAC 2103 Module 10 lnner Product Spaces I...

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