Linear dependence

 

In linear algebra, a set of vectors is said to be linearly dependent if one of the vectors in the set can be expressed as a linear combination of the others. This means that the vectors are not "independent," and that the set is not a basis for the space.

For example, consider the following set of vectors:

{[1 2], [2 4], [4 8]}

These vectors are linearly dependent, because the third vector can be written as a scalar multiple of the first vector: [4 8] = 2[1 2].

On the other hand, consider the following set of vectors:

{[1 2], [3 4]}

These vectors are linearly independent, because it is not possible to write one vector as a linear combination of the others.

Linear dependence is an important concept in linear algebra, because it determines whether a set of vectors can be used as a basis for a space. A basis is a set of linearly independent vectors that span the space, and any vector in the space can be written as a unique linear combination of the basis vectors.