*This preview shows
pages
1–6. Sign up
to
view the full content.*

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **Chapter 1 Vector Spaces Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field F is a set with the operations addition and scalar multiplication , so that for each pair x,y V there is a unique x + y V, and for each a F and x V there is a unique ax V, such that: (VS 1) For all x,y V, x + y = y + x . (VS 2) For all x,y,z V, ( x + y ) + z = x + ( y + z ) . (VS 3) There exists V such that x + = x for each x V. (VS 4) For each x V, there exists y V such that x + y = . (VS 5) For each x V, 1 x = x . (VS 6) For each a,b F and each x V, ( ab ) x = a ( bx ) . (VS 7) For each a F and x,y V, a ( x + y ) = ax + ay . (VS 8) For each a,b F and each x V, ( a + b ) x = ax + bx . n-tuples Example The set of all n-tuples ( a 1 ,a 2 ,...,a n ) with a 1 ,a 2 ,...,a n F is denoted F n . This is a vector space with the operations of coordinatewise addition and scalar multiplication: If c F and u = ( a 1 ,a 2 ,...,a n ) F n , v = ( b 1 ,b 2 ,...,b n ) F n , then u + v = ( a 1 + b 1 ,a 2 + b 2 ,...,a n + b n ) , cu = ( ca 1 ,ca 2 ,...,ca 3 ) . u,v are equal if a i = b i for i = 1 , 2 ,...,n . Vectors in F n can be written as column vectors a 1 a 2 . . . a n or row vectors ( a 1 ,a 2 ,...,a n ) . Matrices Example An m n matrix is an array of the form a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . a m 1 a m 2 a mn where a ij F for 1 i m , 1 j n . The set of all these matrices is denoted M m n ( F ) , which is a vector space with the operations of matrix addition and scalar multiplication : For A,B M m n ( F ) and c F , ( A + B ) ij = A ij + B ij ( cA ) ij = cA ij for 1 i m , 1 j n . Functions and Polynomials Example Let F ( S,F ) denote the set of all functions from a nonempty set S to a field F . This is vector space with the usual operations of addition and scalar multiplication: If f,g F ( S,F ) and c F : ( f + g )( s ) = f ( s ) + g ( s ) , ( cf )( s ) = c [ f ( s )] for each s S....

View
Full
Document