1. 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

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,+,) ]
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

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 trivial space.

7. 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

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, 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+.

12. THANK YOU