ch4 - Vector Space abstraction and generalization of Rn...

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Vector Space : abstraction and generalization of R n . study the implications of the key vector operations: vector addition and scalar multiplication. similar types of operations also exist for various collec- tions of algebraic objects. “dot product” “inner product”. “cross product” “wedge product”, skipped. 1
Definition of a Vector Space A collection of vectors: V , scalars for scaling : R ; A vector addition: +, A scalar multiplication: · for vector addition : (i) (closure of vector addition) u + v is always in V . (ii) (commutative law) u + v = v + u . (iii) (associative law) ( u + v ) + w = u + ( v + w ). (iv) (existence of zero vector) has 0 s.t. u + 0 = u . (v) (existence of negative vector) for each u V , there is a vector - u s.t. u + ( - u ) = 0 . 2
...continued for scalar multiplication : (vi) (closure of scalar multiplication) c u is always in V . (vii) (distributive law) c ( u + v ) = c u + c v . (viii) (distributive law) ( c + d ) u = c u + d u . (ix) (compatibility) c ( d u ) = ( cd ) u . (x) (normalization) 1 u = u . Def : A non-empty set of vectors V with vector addition “+” and scalar multiplication “ · ” satisfying all the above properties is called a vector space over R . 3
Facts : (i) The zero vector 0 so defined is unique. 0 = 0 + w = w (ii) The negative vector for each u is unique. - u = - u + 0 = - u + ( u + w ) = ( - u + u ) + w = 0 + w = w We define vector subtraction u - v := u + ( - v ). (iii) c 0 = 0 . (iv) 0 u = 0 . (v) - u = ( - 1) u . 4
Examples of Common Vector Spaces • { 0 } , zero vector space. R n : ordered n -tuples of real numbers with entry-wise vector addition and scalar multiplication. u + v = u 1 + v 1 . . . u n + v n , c u = cu 1 . . . cu n Blue-print for vector space. 5
S , the doubly infinite sequences of numbers: { y k } = ( . . . , y - 2 , y - 1 , y 0 , y 1 , y 2 , . . . ) with component-wise addition and scalar multiplication. { y k } + { z k } = ( . . . , y - 2 , y - 1 , y 0 , y 1 , y 2 , . . . ) + ( . . . , z - 2 , z - 1 , z 0 , z 1 , z 2 , . . . ) := ( . . . , y - 1 + z - 1 , y 0 + z 0 , y 1 + z 1 , . . . ) c { y k } := ( . . . , cy - 2 , cy - 1 , cy 0 , cy 1 , cy 2 , . . . ) zero vector: { 0 } , sequence of zeros. 6
P n : collection of polynomials with degree at most equal to n and coefficients chosen from R : p ( t ) = p 0 + p 1 t + . . . + p n t n , with polynomial operations. “vector addition”: “ p ( t ) + q ( t )” is defined as: p ( t ) + q ( t ) = ( p 0 + q 0 ) + ( p 1 + q 1 ) t + . . . + ( p n + q n ) t n ; “scalar multiplication”: “ cp ( t )” is defined as: cp ( t ) = ( cp 0 ) + ( cp 1 ) t + . . . + ( cp n ) t n . “zero vector”: the zero polynomial 0( t ) 0. P : all polynomials. 7
The collection of all real-valued functions on a set D , i.e. f : D R . ( D usually is an interval.) “vector addition”: the function “ f + g ” is defined as: ( f + g )( x ) = f ( x ) + g ( x ) for all x in D. “scalar multiplication”: the function “ cf ” is defined as: ( cf )( x ) = cf ( x ) for all x in D. “zero vector”: the zero function 0( x ) which sends every x in D to 0, i.e. 0( x ) = 0 for all x in D. 8
M m × n : collection of all m × n matrices with entries in R with matrix addition and scalar multiplication. A + B = ( a ij + b ij ) , cA = ( ca ij ) . zero vector: m × n zero matrix O m × n . . . . and many more . 9
Subspace : A subspace H of V is a non-empty subset of V such that H itself forms a vector space under the same vector addition and scalar multiplication induced from V . Checking : Following conditions are automatically valid: (ii) u + v = v + u . (iii) ( u + v ) + w = u + ( v + w ). (vi) c ( u + v ) = c u + c v . (viii) ( a + b ) u = a u + b u . (ix) ( ab ) u = a ( b u ). (x) 1 u = u . 10
Need to verify the remainings: (i) the sum u + v is always in H . (iv) there is a zero vector 0 in H . (v) for each u H

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