Page 79 - DMTH502_LINEAR_ALGEBRA

Unit 3: Bases and Dimensions of Vector Spaces

a 0,a 0,...a 0 Notes
1 2 n
Therefore, S is linearly independent set.
Further, we must show that L(S) = V (F).
n
Let v v ,v ,...v be any vector in V (F). We can write
1 2 n n
v ,v ,...v v 1,0,...,0 v 0,1,...,0 ... v 0,0,...,1
1 2 n 1 2 n
i.e., v v e v e ... v e .
2 2
n n
1 1
Hence S is a basis of V (F).
n
Example 7: Prove that the vector space F(x) of polynomials over the field F has a basis S,
such that S 1, ,x 2 ,... .
x
Solution: Let a, b, c, ... be scalars such that
a(1) + b(x) + c(x ) + ... + = 0
2
a 0,b 0,c 0,...
2
the vectors 1, x, x ,... are linearly independent.
2
Let f(x) = a + a x + a x + ... + a x be a polynomial in the given vector space then
i
0 1 2 i
f x a 0 1 a x a x 2 ...a x i
1
2
i
f x can be expressed as a linear combination of a finite number of elements of 1, ,x x 2 ,... .
Thus 1, ,x x 2 ,... is a basis.
Self Assessment

,
1. Find the condition that the vectors a a and b 1 ,b 2 in V (F) are linearly dependent.
1
2
2
2 1 = 0]
a b a b
[Ans: 1 2
2. Test the linear dependence or independence of the vectors:
(i) 0,1, 2 , 1, 1,1 , 1,2,1 in V R
1 2 3 3
(ii) 1,2,3 , 3, 2,1 2, 6,5 in R 3

(iii) 1,0, 1 , 2,1,3 1,0,0 1,0,1 in V R .
3
(iv) The set 1,2,1 , 3,1,5 3,4,7

3. Is the vector 2, 5, 3 in V R ,a linear combination of vectors.
3
1, 3,2 , 2, 4, 1 , 1, 5, 7 ?
1 2 3
4. Prove that the number of elements in a basis of a finite dimensional vector space is unique.

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