Applications of Vector and Matrix Algebra

Abstract

These concepts apply to all kinds of entities in a set {x₁, x₂, x₃ …} as long as the entities can be multiplied by scalars and added. A linear combination of x₁, x₂, x₃, … is :

$$x = \sum\limits_r {{\lambda _r}{x_r}} {\text{ for some scalars }}{\lambda _1},{\lambda _2},{\lambda _3}, \ldots $$

A set {x₁, x₂, x₃, …} is linearly dependent if at least one member is a linear combination of other members. The concepts are particularly important, indeed quite fundamental, for sets of vectors and matrices.