1. Elementary Linear Algebra
4.1 Real Vector Spaces
4.2 Subspaces
4.3 Linear Independence
4.4 Coordinates and Basis
4.5 Dimension
4.6 Change of Basis
4.7 Row Space, Column Space, and Null Space
4.8 Rank, Nullity, and the Fundamental Matrix Spaces

2. Definition 1.1: Definition: Let V be an arbitrary non empty set of objects on which two operations are defined (1) addition and (2) multiplication by scalars ( 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

3. To Show that a Set with Two Operations is a Vector Space
1. Identify the set V of objects that will become vectors. 2. Identify the addition and scalar multiplication operations on V. 3. Verify Axioms 1(closure under addition) and 6 (closure under scalar multiplication) ; that is, adding two vectors in V produces a vector in V, and multiplying a vector in V by a scalar also produces a vector in V. 4. Confirm that Axioms 2,3,4,5,7,8,9 and 10 hold.

4. Examples
1. The set V of real and complex numbers.
The set of polynomials of order n.
The set of all matrix of order m*n

5. Counter Examples

6. Example : R2
R2 is a vector space if
with
Example : Plane in R3.
The plane through the origin
is a vector space.

7. Example : 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 : 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 : Solution Space of a Linear Homogeneous Differential Equation
is a vector space with
Vector addition:
Scalar multiplication:
Zero element:
Inverse:
Closure: →

15. Rotation Operators

17. Dilations and Contractions

19. Expansion or Compression

20. Shear