Page 77 - DMTH502_LINEAR_ALGEBRA

Unit 3: Bases and Dimensions of Vector Spaces

Notes
0, 0, 0.
2
Hence the vectors 1, ,1x x x are linear independent over the field of real numbers.

Example 4: If the set S 1 , 2 ,... n of vectors ofV F is linearly independent, then
none of the vectors 1 , 2 ,... n can be zero vector.

Solution: Let be zero vector where 1 r n then
r
0 0 ... a 0 r ... 1 0 0
1 2 r n
for any a 0 in .
F
Since a 0 we notice that S is linearly dependent. This is contrary to what is given.
Hence none of the vectors , ,... can be a zero vector.
1 2 n
3.2 Basis and Dimension of a Vector Space

A subset S of a vector space V F is said to be a basis of V F , if
(i) S consists of linearly independent vectors, and

(ii) S generates V F i.e. ( )S V i.e. each vector in V is a linear combination of the finite
number of elements of S.
For example the set (1, 0, 0), (0, 1, 0), (0, 0, 1) is a basis of the vector space V (R) over the field of
3
real numbers.
The set = (v , v , v , …, v ) is a basis of V if every vector w in V can be written in a unique way
1 2 3 n
as a combination w = x v + x v + ……………… + x v .
1 1 2 2 n n
If every vector can be uniquely written as a combination, of the vectors v , v , … v of , then
1 2 x
is independent and spans V, so is a basis.
If V is a finite dimensional vector space, then it contains a finite set v , v , …, v of linearly
1 2 n
independent elements that spans V.
If v , v , … v is a basis of V over F and if w , w , … w in V are linearly independent over F, then
1 2 n 1 2 m
m n.
We also see that if V is finite-dimensional over F then any two basis of V has the same number
of elements.
Thus for a finite dimensional space V, the basis has a unique number of elements say n. This
unique integer, n; in fact, is the number of elements in any basis of V over F.
Definition: The integer n is called the dimension of the vector space over F.
The Dimension of a finite space V over F is thus the number of elements in any basis of V over
F.
A vector space V is finite-dimensional if some finite set of vectors spans V. Otherwise V is
infinite dimensional.
The dimension of V will be denoted by dim V.
If W is the subspace of a finite dimensional vector space V, then W is finite dimensional, and
dim W dim V. Moreover, dim W = dim V if and only if W = V.

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