r r Prof. Inder K. Rana Room 112 B Department of Mathematics - PowerPoint PPT Presentation

▲✐♥❡❛r ❆❧❣❡❜r❛ Prof. Inder K. Rana Room 112 B Department of Mathematics IIT-Bombay, Mumbai-400076 (India) Email: ikr@math.iitb.ac.in Lecture 13 Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Abstract vector spaces R 3 The algebraic structures of addition and scalar multiplication on I generalized to the notion of Vector Spaces that arise in many other branches of mathematics and other disciplines. Definition (Abstract vector space) Let V be a non-empty set and I F “ I R p or I C q . Let V have two algebraic operations: VECTOR ADDITION: 1 ` : V ˆ V Ý Ñ V , p a , b q ÞÑ a ` b and SCALAR MULTIPLICATION: 2 p¨q : I F ˆ V Ý Ñ V , p λ, a q ÞÑ λ a is called a vector space if the following eight axioms hold: Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Axioms contd. @ a , b , c P V and λ P Definition (continuation) A1 a ` b “ b ` a . [Commutativity] A2 p a ` b q ` c “ a ` p b ` c q . [Associativity] A3 D ! 0 P V s.t. a ` 0 “ a . [Additive identity] A4 D ! ´ a s.t. a ` p´ a q “ 0 . [Additive inverse] M1 λ p a ` b q “ λ a ` λ b . [Distributivity(scal. mult. over vect. add.)] M2 p λ ` µ q a “ λ a ` µ a [Distributivity(scal. mult. over scal. add.)] M3 λ p µ a q “ p λµ q a [mixed associativity] M4 and 1 ¨ a “ a [normalization] Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Some natural consequences The following natural properties can be logically deduced form the above axioms. 0 ¨ a “ 0 “ λ ¨ 0 . p´ 1 q a “ ´ a . Remark: The uniqueness assertion in A 3 and A 4 is also derivable if we chose to omit it from the axioms. i.e. write D instead of D ! Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Examples Example (Vector spaces) R n and I C n . I 1 R n | A x “ 0 u , A P M m ˆ n p I t x P I R q . 2 M m ˆ n p I R q and M m ˆ n p I C q . 3 R 3 r x s , the set of all the polynomials in x with real coefficients of I 4 degree ď 3. C 3 r x s or I R d r x s , etc., can be defined. Similarly I R r x s , p I C r x sq , the set of all the polynomials in variable x with real 5 I (complex) coefficients. Solutions of the equation y 2 ` µ 2 y “ 0 or of the equation 6 y 1 ` q p x q y “ 0 (over some interval I ). „ y 1 p t q  satisfying : The set of vector functions y p t q “ y “ A y , where 7 y 2 p t q A P M 2 p I R q . Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Non-examples Example (Non vector spaces) All m ˆ n real matrices with entries ě 0. 1 Solutions of xy 1 ` y “ 3 x 2 . 2 Solutions of y 1 ` y 2 “ 0. 3 Definition Let V be a vector space. A subset W of V is called a subspace of V if for w 1 , w 2 P W and α, β P I F , α w 1 ` β w 2 P W . Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Linear combinations Definition (Linear combinations) Given v 1 , v 2 , ..., v k P V , a linear combination is a vector c 1 v 1 ` c 2 v 2 ` ¨ ¨ ¨ ` c k v k for any choice of scalars c 1 , c 2 , ..., c k . Definition Let S be a nonempty subset of V . Let �ř n ( r S s : “ i “ 1 α i v i | n P I N ; α 1 , ...α n P I F ; v 1 , ... v n P S . It is called the subspace generated by S . Definition Let V be a vector space such that there exists a finite set S Ă V with r S s “ V . In such a case we say that V is a finitely generated vector space. Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Examples Finitely generated Vector spaces R n and I C n , I 1 R n | A x “ 0 u , A P M m ˆ n p I N p A q : “ t x P I R q . 2 M m ˆ n p I R q and M m ˆ n p I C q . 3 R 3 r x s -the set of all the polynomials in x with real coefficients of I 4 degree ď 3. Similarly I C 3 r x s or I R d r x s can be defined. Vector spaces which is not finitely generated R r x s -the set of all the polynomials in x with real coefficients. Similarly I C r x s . I Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Linear combinations Definition (Linear dependence) A set of vectors v 1 , v 2 , ..., v k P V is called linearly dependent If scalars c 1 , c 2 , ..., c k , at least one non zero, can be found such that c 1 v 1 ` c 2 v 2 ` ¨ ¨ ¨ ` c k v k “ 0 . Contrapositive of linear dependence is linear independence Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Example Example R r x s the set t 1 , x , x 2 , x 5 u is a linearly independent set: In I 1 If c 1 ` c 2 x ` c 3 x 2 ` c 4 x 5 “ 0 , right hand side being the zero polynomial, implies, by equating coefficients of like powers, c 1 “ c 2 “ c 3 “ c 4 “ 0 . On the other hand the set t 1 , x , 1 ` x 2 , 1 ´ x 2 u Ă I R r x s is a linearly 2 dependent set since p´ 2 q ˆ 1 ` 0 ˆ x ` 1 ˆ p 1 ` x 2 q ` 1 ˆ p 1 ´ x 2 q “ 0 , implying there is a linear combination which is zero but not all the scalers are zero. Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Basis Definition (Basis) If a set of vectors S in a vector space V is such that S is linearly independent and Every vector in V is a (finite) linear combination of vectors from S . then the set S is called a basis of S . Theorem (Existence of basis) Every vector space V has a basis. 1 If V is finitely generated, then any two basis will have same 2 number of elements. The number of elements in any basis of V is called the dimension of V . If V is not finitely generated, we say it is infinite dimensional . 3 Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

More examples Example R 5 r x s is S “ t 1 , x , x 2 , x 3 , x 4 , x 5 u . A basis of I 1 For this note that any polynomial p P I R 5 r x s is p p x q “ c 0 ` c 1 x 1 ` c 2 x 2 ` c 3 x 3 ` c 4 x 4 ` c 5 x 5 ` c 6 x 6 and c 0 ` c 1 x 1 ` c 2 x 2 ` c 3 x 3 ` c 4 x 4 ` c 5 x 5 ` c 6 x 6 “ 0 implies each c i “ 0 . Hence S is a basis of I R 5 r x s , which says that it is 6-dimensional. is Consider S “ t 1 , x , x 2 , ..., x n , ... u Ă I R r x s . It is easy to check that 2 S is a linearly independent set. Hence I R r x s is infinite R r x s . dimensional. In fact S forms a basis of I Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

More examples Example R 3 . t i , j , k u is the standard basis of I R n . t e 1 , e 2 , ..., e n u is the standard basis of I t E jk , 1 ď j ď m , 1 ď k ď n u Ă M m ˆ n p I R qu is a basis of M m ˆ n p I R q . t cos µ x , sin µ x u is a basis of the solution space of the differential equation y 2 ` µ 2 x “ 0 . ˇ Let V “ t p p x q P I R 3 r x s ˇ p p 1 q “ 0 u . Then V is a vector space. It has B “ t 1 ´ x , 1 ´ x 2 , 1 ´ x 3 u as a basis. It has 3 dimensions. Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Description of finite dimensional vector spaces Let Let B : “ t v 1 , . . . , v n u be any ordered basis of a vector space V over F of dimension n .For v P V , if α 1 , α 2 , . . . , α n P I I R are the unique scalars such that v “ α 1 v 1 ` α 2 v 2 ` . . . ` α n v n , then we write » fi α 1 . . r v s B : “ — ffi . – fl α n and call it the coordinate vector of v . This gives us an identification F n . v ÞÑ r v s B from V to I It is easy to show that (i) r u 1 ` u 2 s B “ r u 1 s B ` r u 2 s B . (ii) r λ u s B “ λ r u s B . F n . Thus for all practical purposes, V – I Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Linear maps on vector spaces Definition (Linear map or transformation) Let V , W be two (abstract) vector spaces. A map T : V Ý Ñ W is called a linear map or transformation if T p v ` w q “ T v ` T w and T p λ v q “ λ T v . Example R n Ý R m via the A P M m ˆ n p I R q can be viewed as a linear map A : I Ñ I matrix multiplication v ÞÑ A v . Here v is treated as n ˆ 1 column which maps to m ˆ 1 column A v . Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

More examples d (a) R r x s Ý Ñ I R r x s dx : I ` ˘ R r x s m Ý Ñ I R r x s m where τ c p p x q “ p p x ´ c q for any (b) τ c : I scalar c P I R . ` ˘ (c) µ x : I R r x s m Ý Ñ I R r x s defined by µ x p p x q “ xp p x q . (d) I : C pr 0 , 1 sq Ý Ñ C pr 0 , 1 sq defined by ż x ` ˘ I p f q p x q “ f p t q dt , 0 ď x ď 1. 0 Levels of complexity of maps: Constant maps. Easiest to describe. Need to know at any one 1 point only. Linear maps. Easiest among non constant maps. Enough to 2 sample on a basis to determine it completely. (Constant maps can not describe dependence of output on input.) Non-linear maps. Hardest to deal with. Often the local behaviour 3 is studied by linear approximation . Hence the importance of linear maps. Prof. Inder K. Rana Department of Mathematics, IIT - Bombay

Matrix of a linear transformation) Let T : V Ý Ñ W is a linear map, where V and W are finite dimensional vector spaces with ordered bases B 1 “ t v 1 , v 2 , ..., v n u and B 2 “ t w 1 , w 2 , ..., w m u of V and W ,respectively. Let T v 1 “ t 11 w 1 ` t 21 w 2 ` ¨ ¨ ¨ ` t m 1 w m “ t 12 w 1 ` t 22 w 2 ` ¨ ¨ ¨ ` t m 2 w m T v 2 . . . . . . . . . T v n “ t 1 n w 1 ` t 2 n w 2 ` ¨ ¨ ¨ ` t mn w m This gives us a (unique) matrix representation r t jk s of T w.r.t. to the ordered bases B 1 and B 2 of V and W respectively. Prof. Inder K. Rana Department of Mathematics, IIT - Bombay