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.

Linearly Dependent Vectors


Properties

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

Linearly Dependent Vectors

Linearly Dependent Vectors

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 vector u = (u1, u2, u3) y v = (v1, v2, v3) are linearly dependent if their components are proportional.

Linearly Dependent Vectors

Vectors

Vector Matrix


Example

Determine the values of k for the linearly dependent vectors Vector U, Vector V. and Vector W. Also, write Vector U as a linear combination of Vector V and Vector W, 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.

Determinant of the Matrix

Linearly Dependent Vector Calculations

Linearly Dependent Vector Calculations

Linearly Dependent Vector Calculations

Linearly Dependent Vector Calculations

System of Equations

Linearly Dependent Vector Solution