Linear dependence of vectors

In this publication, we will consider what a linear combination of vectors is, which vectors are linearly dependent and independent, and also analyze an example of a problem on this topic.

Determining the linear dependence of vectors

Linear combination of vectors a₁,…, a_n is the vector given by the expression x₁a₁ + … + x_na_nWhere x₁,…, X_n – coefficients.

The presented combination can be:

• Trivial – all odds x₁,…, X_n equal zero.
• Non-trivial at least one of the coefficients x₁,…, X_n is not equal to zero.

Vectors a₁,…, a_n linearly independent, if only their trivial combination is equal to the zero vector. That is:

a₁,…, a_n are linearly independent if x₁a₁ + … + x_na_n = 0, only when x₁ = 0, …, x_n = 0.

Vectors a₁,…, a_n linearly dependent, if there is such a non-trivial combination of them that is equal to the zero vector.

Properties of linearly dependent vectors

• Linearly Dependent Vectors in Two/Three Dimensional Space. The converse is also true.
• In XNUMXD space, there are three linearly dependent vectors . The statement is also true in reverse.

Example of a problem

Let’s check if the vectors are a = {1; 2} и b = {2; 4} linearly dependent.

Decision:

We need to find the values ​​of the coefficients for which the linear combination of these vectors equals the zero vector.

xa + Yb = 0

The resulting vector equation can be represented as a system of linear equations:

This system has many solutions, while x = -2y.

That is, there is a non-zero combination of coefficients x и y, for which the combination of vectors a и b equals the null vector. Therefore, the given vectors are linearly dependent.