RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE, HADAPSAR, PUNE-411028

1. RAYAT SHIKSHAN SANSTHA’SS.M.JOSHI COLLEGE, HADAPSAR, PUNE-411028 PRESANTATION BY Prof . DESAI S.S Mathematics department Subject – Linear Algebra Topic - Vector Spaces and Subspaces

2. I. Definition of Vector Space I.1. Definition and Examples I.2. Subspaces

3. I.1. Definition and Examples • Definition 1.1: (Real) Vector Space ( V,  ; R ) • A vector space(over R) consists of a set V along with 2 operations ‘’ and ‘’ s.t. • For the vector addition : •  v, w, u V • v w V ( Closure ) • v w = w v ( Commutativity ) • ( v w) u = v  ( wu ) ( Associativity ) •  0 V s.t. v0 = v ( Zero element ) •  v V s.t. v  (v) = 0 ( Inverse ) • (2) For the scalar multiplication : •  v, w V and a, b R, [ R is the real number field (R,+,) • a  v V ( Closure ) • ( a + b ) v = ( a v )  (b  v ) ( Distributivity ) • a ( v  w ) = ( a  v )  ( a  w ) • ( a  b ) v = a ( bv ) ( Associativity ) • 1 v = v •  is always written as + so that one writes v + w instead of v w •  and  are often omitted so that one writes a b v instead of ( a  b )  v

4. Definition in Conventional Notations • Definition 1.1: (Real) Vector Space ( V, + ; R ) • A vector space(over R) consists of a set V along with 2 operations ‘+’ and ‘’ s.t. • For the vector addition+ : •  v, w, u V • v +w V ( Closure ) • v +w = w +v ( Commutativity ) • ( v +w) +u = v + ( w+u ) ( Associativity ) •  0 V s.t. v+0 = v ( Zero element ) •  v V s.t. v v = 0 ( Inverse ) • (2) For the scalar multiplication : •  v, w V and a, b R, [ R is the real number field (R,+,) ] • av V ( Closure ) • ( a + b ) v = av +b v ( Distributivity ) • a ( v + w ) = av + a w • ( a  b ) v = a ( bv )= abv( Associativity ) • 1 v = v

5. Example 1.3: R2 R2 is a vector space if with Proof it yourself / see Hefferon, p.81. Example 1.4: Plane in R3. The plane through the origin is a vector space. P is a subspace of R3. Proof it yourself / see Hefferon, p.82.

6. Example 1.5: Let  &  be the (column) matrix addition & scalar multiplication, resp., then ( Zn, + ; Z ) is a vector space. ( Zn, + ; R ) is not a vector space since closure is violated under scalar multiplication. Example 1.6: Let then (V, + ; R ) is a vector space. Definition 1.7: A one-element vector space is a trivialspace.

7. Example 1.8: Space of Real Polynomials of Degree n or less, Pn E.g., The kth component of a is Pn is a vector space with vectors Vector addition: i.e., Scalar multiplication: i.e., Zero element: i.e., Inverse: i.e., Pn is isomorphic to Rn+1 with

8. Example 1.9: Function Space The set { f | f : N → R } of all real valued functions of natural numbers is a vector space if Vector addition: Scalar multiplication: Zero element: Inverse: f ( n ) is a vector of countably infinite dimensions: f = ( f(0), f(1), f(2), f(3), … ) E.g., ~

9. Example 1.10: Space of All Real Polynomials, P P is a vector space of countably infinite dimensions. Example 1.11: Function Space The set { f | f : R → R } of all real valued functions of real numbers is a vector space of uncountably infinite dimensions.

10. Example : The x-axis in Rn is a subspace. Proof follows directly from the fact that • Example : • { 0 } is a trivial subspace of Rn. • Rn is a subspace of Rn. • Both are improper subspaces. • All other subspaces are proper. Example : Subspace is only defined wrt inherited operations. ({1}, ; R) is a vector space if we define 11 = 1 and a1=1 aR. However, neither ({1}, ; R) nor ({1},+ ; R) is a subspace of the vector space (R,+ ; R).

11. Example : Polynomial Spaces. Pn is a proper subspace of Pm if n < m. Example : Solution Spaces. The solution space of any real linear homogeneous ordinary differential equation, Lf = 0, is a subspace of the function space of 1 variable { f : R → R }. Example : Violation of Closure. R+ is not a subspace of R since (1) v  R+  v R+.

12. THANK YOU