Lecture Notes 7

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online