Linear Dependence

Definition:

If  is a set of vectors, then the equation

has a solution .

If this is the only solution, then A is linearly independent. If there are other solutions, then A is linearly dependent. Determining linear dependence is important to performing other operations of linear algebra.

Example 1:

to determine linear dependence we solve:

therefore:

solving for constants: (this can be accomplished using inforation provided in the solving section)

(K₁, K₂, K₃ )=(2, 1, 3)

This is a non-trivial solution so the set is linearly dependent.

Example 2:

to determine linear dependence we solve:

K₁(1,0,0)  + K₂(0,0,2) + K₃(0,1,0) = (0,0,0)

therefore:

K₁=0, 2K₃=0, K₂=0

solving for constants:

This is the trivial solution, so the set is linearly independent.

Graphical Representation:

The following are two-dimentional examples of linearly dependent and linearly independent vectors;

U₁=(1,5) U₂=(3,2)

(K₁)U₁+(K₂)U₂=0

we graph K₁ vs K₂ for the two vector components     

K₁    

K₂   

The two graphs intersect only at the orgin so the vectors are linearly independent.

For the the following vectors;

U₁=(1,5) U₂=(2,10)

(K₁)U₁+(K₂)U₂=0

we graph K₁ vs K₂ for the two vector components     

K₁ 

K₂

The two graphs share an infinite number of points because they lie on the same line. Therefore, the vectors are linearly dependent because there are infinite solutions to the system of equations.