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

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

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

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,+,) ] 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 ) = a b v ( Associativity ) 1 v = v

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.

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 trivial space.

Example 1.8: Space of Real Polynomials of Degree n or less, Pn 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

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., ~

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.

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).

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, L f = 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+.

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