# Linearly Independent Vectors

Several vectors are linearly independent if none of them can be expressed as a linear combination of others.

**a _{1} = a_{2} = ··· = a_{n} = 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.