1. Vector Spaces
A set V is called a vector space over a set K denoted V(K) if
is an Abelian group,
is a field,
and
For every element vV and K there exists an element .v V called the “scalar multiple of v by ” satisfying
(i)
(ii)
(iii)
(iv)
Notation : 0 K denotes the additive identity under +, K, - denotes the inverse of  under +
denotes the identity under
v denotes the inverse of v under

2. Examples Example 1 (Polynomials of degree n)
V = set of all polynomials of order n
The additive operation on vectors is defined as follows:
V(K) is a vector space
Proof
For v=
Now
by continuity of addition
on the real numbers

3. Examples:1 Proof continued follows from associativity of
normal addition
Closure is trivial
Hence,
is an Abelian group
therefore
Similarly for properties (ii)-(iv)

4. Examples:2 Example 2 (n dimensional vectors)
Example 3 (Complex Numbers)
Example 4 (Matrices)
is the set of nm matrices
is the set of nn matrices

5. Properties of Vector Spaces
Theorem
Proof
Identity under +
by axiom (ii) of vector spaces
But
Identity under 
Therefore
by the cancellation law for V K v

6. Properties:2 Theorem (i) = .( v) (ii) (-).( v) =.v Proof (i) Show
by previous theorem and inverse under +
Therefore,
by axiom (ii) of vector spaces
Also
(.v)
by inverse under 
Therefore
(.v)
(.v)
by the cancellation law for

7. Properties:3 Show = .( v) .(v  ( v)) by inverse under  Therefore,
by axiom (i) vector spaces
Now
by previous theorem and axiom (iii)
by above theorem
.v  . v
Also
.v 
(.v)
by inverse under 
Therefore,
.v  . v=
.v 
(.v)
. v=
(.v)
By the cancellation law for
(ii) proof ??

8. Subspaces
Definition
Let WV such that W then W(K) is a subspace of V(K) if W is a vector space over K with the same definition of  and scalar multiple as V
Clearly to show that W(K) is a subspace of V(K) we need only show that <W,> is a sub-group of <V, > and that
Characterisation Theorem
A non-empty subset W of V is a subspace of V iff .u  vW for all K, u,vW
Proof
() trivial since .uW

9. Subspaces:2 Proof (continued) () Taking =1 then  u,vW
by axiom (iv)
Taking =-1 then  u,vW
and by a previous theorem
(-1).u=
(1.u)= u
by axiom (iv)
Therefore
u  uW 
by inverse under 
Therefore K, uW taking
gives .uW
Therefore, taking =-1 gives 
(1.u)W 
uW
by a previous theorem
Hence,
is a subgroup and
as required

10. Examples of Subspaces Example 1 Let Then W(K) is a subspace of V(K)

11. Examples of Subspaces:2
Example 1 (continued)
proof
(i) Clearly W and WV
(ii) For R and u,vW such that
and
then
Example 2
(i) Clearly W and WV
and 2R but
W(K) is not a subspace of V(K)

12. Linear Combinations Definition If then where
is a linear combination of S

13. Linear Combinations:2 Theorem For then is a subspace of V Proof since
and
by a previous theorem and hence If
then
and
for some
Then
by axiom (i)
by axiom (iii)
by axiom (ii)

14. Linear Independence Definition A subset of V is linearly independent
iff
Otherwise if there exist one
such that
then
are linearly dependent
Example 1
is linearly dependent over R since
Example 2
is linearly independent since

15. Linear Dependence in Matrices
Theorem If
and
are the n column vectors of A then
is linearly dependent over K if and
only if
Proof If
is linearly dependent over K, then
there exists
(not all zero)
such that
Without loss of generality assume
then
Hence, performing a column operation where
is added to column 1 gives
a matrix with zero first column. Hence,

16. Matrices:2 Proof (continued) If then the system of equations
has a non-trivial solution,
But this is the same as saying that there exist
(not all zero) such that
By considering the transpose of A we obtain
Corollary
The n rows of a matrix
are linearly
dependent if and only if

17. Basis Definition A set is a basis for V iff
(i) S is linearly independent over K
(ii)
Condition (ii) means that S is a spanning set for V
Definition (Finitely Generated)
A vector space V is said to be finitely generated if it has a Basis S with a finite number of elements V S

18. Examples of Basis Example 1 Let is a basis for V Then
Linear independence:
Spanning:
is also a basis for V
Proof ??

19. Examples of Basis:2 Example 2 Let then is a basis where Example 3 then

20. Dimension
Theorem
Every Basis of a finitely generated vector space has the same number of elements.
Definition (Dimension)
The number of elements in a basis for a finitely generated vector space V is called the dimension of V and denoted dim V.
Examples
then
is a basis
dim(V) = 3
Let
then
is a basis and dim(V)=4