Vector Spaces,subspaces,Span,Basis

Vector Calculs

1. VCLA (2110015) Active Learning Assigment Branch-IT Div:- D_DG1 Group Members Ravi Gelani (150120116020) Simran G hai (150120116021) Topic :-Vector spaces , subspaces , span , basis Guided by:- Prof.sikha yadav

2. Vector space is a system consisting of a set of generalized vectors and a field of scalars,having the same rules for vector addition and scalar multiplication as physical vectors and scalars. What is Vector Space? Let V be a non empty set of objects on which the operations of addition and multiplication by scalars are defined. If the following axioms are satisfied by all objects u,v,w in V and all scalars k1,k2 then V is called a vector space and the objects in V are called vectors.

3. If u and v are objects in V, then u + v is in V u + v = v + u u + ( v + w ) = ( u + v ) + w There is an object in V, called a zero vector for V, such that + u = u + = u for all u in V For each u in V, there is an object –u in V, called a negative of u , such that u + ( -u ) = ( -u ) + u = If k is any scalar and u is any object in V then k u is in V k ( u + v ) = k u + k v ( k + l )( u ) = k u + l u k ( l u ) = ( kl ) u 1 u = u Addition conditions:-

4. 4 Definition: : a vector space : a non empty subset ： a vector space (under the operations of addition and scalar multiplication defined in V ) W is a subspace of V Subspaces If W is a set of one or more vectors in a vector space V, then W is a sub space of V if and only if the following condition hold; a)If u,v are vectors in a W then u+v is in a W. b)If k is any scalar and u is any vector In a W then ku is in W.

5. 5 Every vector space V has at least two subspaces Zero vector space { } is a subspace of V. (2) V is a subspace of V. Ex: Subspace of R 2 Ex : Subspace of R 3 If w 1 ,w 2 ,. . .. w r subspaces of vector space V then the intersection is this subspaces is also subspace of V .

6. Example: Set Is Not A Vector

7. Span of set of vectors If S ={ v 1 , v 2 ,…, v k } is a set of vectors in a vector space V , then the span of S is the set of all linear combinations of the vectors in S . If every vector in a given vector space can be written as a linear combination of vectors in a given set S , then S is called a spanning set of the vector space. Definition:

8. Notes : span ( S ) is a subspace of V . span ( S ) is the smallest subspace of V that contains S . (Every other subspace of V that contains S must contain span ( S ). If S ={ v 1 , v 2 ,…, v k } is a set of vectors in a vector space V , then

9. Basis Definition:  S is called a basis for V (1) Ø is a basis for { } (2) the standard basis for R 3 : { i , j , k } i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) Notes: S spans V (i.e., span ( S ) = V ) S is linearly independent The set of vectors S ={ v 1 , v 2 , … , v n }  V in vector space V is called a basis for V if ..

10. 10 (3) the standard basis for R n : { e 1 , e 2 , …, e n } e 1 =(1,0,…,0), e 2 =(0,1,…,0), e n =(0,0,…,1) Ex: R 4 {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)} Ex : matrix space : (4) the standard basis for m  n matrix space: { E ij | 1  i  m , 1 j  n } (5) the standard basis for P n ( x ): {1, x , x 2 , …, x n } Ex: P 3 ( x ) {1, x , x 2 , x 3 }

11. Thank you………..