What is a linear combination?

A linear combination is a mathematical concept that involves multiplying a set of numbers by specific coefficients and then adding them together. In other words, it is a combination of variables or terms that are added or subtracted with coefficients.

For example, given the variables a, b, and c, a linear combination of these variables would be expressed as:

αa + βb + γc

Where α, β, and γ are coefficients representing the weights or proportions of each variable in the combination.

Linear combinations are commonly used in various fields of mathematics, including linear algebra, functional analysis, and statistics. They provide a way to express complex relationships between variables in a straightforward and concise manner. Additionally, linear combinations play a key role in solving systems of linear equations and in studying vectors and vector spaces.