Lecture Notes 7

Lecture Notes 7 - Untitled Lecture 07 Vector Spaces and...

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Untitled 1 September 25, 2007 Outline: 1) Intro: toward a deeper understanding of Ax =b 2) Vector Spaces Definition and Rules Real Vector Spaces R n Other Vector Spaces (C n ,R m x n ,Z,F) 3) Vector Subspaces Definition Lots of Examples 4) Fundamental Subspaces associated with a matrix A The Column Space C(A) The Null Space N(A) Lecture 07: Vector Spaces and Subspaces #1 Linear Vector Spaces: an abstract definition Formal Definition: A vector space is a set of objects (vectors) and rules of vector addition and scalar multiplication that satisfy the following axioms 1) x + y = y +x 2) x + (y + z ) = (x +y ) +z 3) There exists a unique zero vector such that x +0 =x 4) For every vector x there is a unique vector -x s.t. x + (-x ) = 0 5) 1 times x equals x 6) (c 1 c 2 )x =c 1 (c 2 x ) 7) c(x + y ) = cx + cy 8) (c 1 + c 2 )x = c 1 x + c 2 x The important consequence of these axioms is that the set is closed under vector addition and scalar multiplication (i.e. any linear combination of vectors chosen from the space remains a member of the space) Linear Vector Spaces: a concrete example the set of real vectors R
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Lecture Notes 7 - Untitled Lecture 07 Vector Spaces and...

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