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.

Linearly Independent Vectors

Examples

1.Determine whether the vectors are linearly dependent or independent.

Vector U = (2, 3, 1), Vector V = (1, 0, 1), Vector W = (0, 3, −1)

a (2, 3, 1) + b(1, 0, 1) + c (0, 3, −1) = (0, 0, 0)

System of Equations

r = 2 n = 3 Consistent dependent system.

The system has infinite solutions, so the vectors are linearly dependent.


2.Demonstrate that Vector U = (1, 0, 1), Vector V = (1, 1, 0) and Vector W = (0, 1, 1) are linearly independent vectors and express the vector Vector M = (1, 2, 3) as a linear combination of these vectors.

Linear Combination

Linear Combination

Linear Combination

System of Equations

The system supports only the trivial solution:

System Solution

Therefore, the three vectors are linearly independent.

Linear Combination

Linear Combination

Linearly Independent Vector Calculations

System of Equations

System of Equations

Linearly Independent Vector Calculations

Linearly Independent Vector Solution