Linearly Independent Vectors
Several vectors are linearly independent if none of them can be expressed as a linear combination of others.
a1 = a2 = ··· = an = 0
Linearly independent vectors have different directions and its components are not proportional.
Examples
1.Determine whether the vectors are linearly dependent or independent.
= (2, 3, 1), 
= (1, 0, 1), 
= (0, 3, −1)
a (2, 3, 1) + b(1, 0, 1) + c (0, 3, −1) = (0, 0, 0)
r = 2 n = 3 Consistent dependent system.
The system has infinite solutions, so the vectors are linearly dependent.
2.Demonstrate that = (1, 0, 1), = (1, 1, 0) and = (0, 1, 1) are linearly independent vectors and express the vector 
= (1, 2, 3) as a linear combination of these vectors.
The system supports only the trivial solution:
Therefore, the three vectors are linearly independent.
