Page 76 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes The following are easy consequences of the definition:
1. Any set which contains a linearly dependent set is linearly dependent.
2. Any subset of linearly independent set is linearly independent.
3. Any set which contains 0 vector is linearly dependent.

4. A set S of vectors is linearly independent if and only if each finite subset of S is linearly
independent.
An infinite subset S of V is said to be linearly independent if every finite subset S is linearly
independent, otherwise it is linearly dependent.

Illustrative Examples

Example 1: Show that the system of three vectors (1, 3, 2)(1, –7, –8), (2, 1, –1) of V R is
3
linearly dependent.
Solution: For 1 , 2 , 3 R such that

1,3,2 1, 7, 8 2,1, 1 0
1 2 3
2 ,3 7 ,2 8 0
1 2 3 1 2 3 1 2 3
2 0,3 7 0,2 8 0
1 2 3 1 2 3 1 2 3
3, 1, 2.
1 2 3
Therefore, the given system of vectors is linearly dependent.

3
Example 2: Consider the vector space R R and the subset S 1,0,0 , 0,1,0 , 0,0,1 of
3
R . Prove that S is linearly independent.
Solution: For 1 , 2 , 3 , R

1,0,0 0,1,0 0,0,1 0,0,0
1 2 3
, , 0,0,0
1 2 3
0, 0, 0.
1 2 3
This shows that if any linear combination of elements of S is zero then the coefficients must be
zero. S is linearly independent.

2
Example 3: Show that 1, ,1x x x is a linearly independent set of vectors in the
vector space of all polynomials over the real number field.
Solution: Let , , be scalar (real numbers) such that
1 x 1 x x 2 0 then

x 1 x x 2 0

x x 2 0
0, 0, 0,

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