# Linearly Dependent Vectors

Vectors are linearly dependent if there is a linear combination of them that equals the zero vector, without the coefficients of the linear combination being zero.

### Properties

1.If several vectors are linearly dependent, then at least one of them can be expressed as a linear combination of the others.

If a vector is a linear combination of others, then all the vectors are linearly dependent.

2.Two vectors are linearly dependent if, and only if they are parallel.

3.Two vectors = (u_{1}, u_{2}, u_{3}) y = (v_{1}, v_{2}, v_{3}) are linearly dependent if their components are proportional.

#### Example

Determine the values of k for the linearly dependent vectors , and . Also, write as a linear combination of and , where k is the calculated value.

The vectors are linearly dependent if the determinant of the matrix is zero, meaning that the rank of the matrix is less than 3.