What is a linear combination of vectors?

A linear combination of vectors is a mathematical operation that involves multiplying each vector by a scalar and then adding the results together. The result is a new vector that lies in the same vector space as the original vectors.

For example, given two vectors v1 = (1, 2) and v2 = (3, 4), a linear combination of these vectors could be written as c1v1 + c2v2 where c1 and c2 are scalars. This would result in a new vector (c11 + c23, c12 + c24) = (c1 + 3c2, 2c1 + 4c2).

The coefficients c1 and c2 determine the weights of each vector in the linear combination. If all the coefficients are zero, then the resulting vector is the zero vector. Otherwise, the linear combination will create a new vector that may lie in a different direction or have a different magnitude than the original vectors.

Linear combinations of vectors are used in various fields such as linear algebra, physics, and engineering to analyze and solve systems of equations, represent transformations, and model real-world scenarios.