{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ch1 - Chapter 1 Vector Spaces Per-Olof Persson...

This preview shows pages 1–7. Sign up to view the full content.

Chapter 1 – Vector Spaces Per-Olof Persson [email protected] Department of Mathematics University of California, Berkeley Math 110 Linear Algebra

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

View Full Document
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 0 V such that x + 0 = x for each x V. (VS 4) For each x V, there exists y V such that x + y = 0 . (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 ) .

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

View Full Document
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. Example The set P ( F ) of all polynomials with coefficients in F : f ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 1 x + a 0 is a vector space with the usual operations of addition and scalar multiplication (set higher coefficients to zero if different degrees): f ( x ) + g ( x ) = ( a n + b n ) x n + · · · + ( a 1 + b 1 ) x + ( a 0 + b 0 ) , cf ( x ) = ca n x n + · · · + ca 1 x + ca 0 .

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

View Full Document
Properties of Vector Spaces Theorem 1.1 (Cancellation Law for Vector Addition) If
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

ch1 - Chapter 1 Vector Spaces Per-Olof Persson...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online