What is a linear combination of vectors?

A linear combination of vectors is defined as the sum of scalar multiples of those vectors. In simpler terms, if you have a set of vectors, you can multiply each vector by a scalar (which is just a constant number) and then sum all those results together. This process allows for the construction of new vectors from the existing ones, emphasizing how linear combinations span a vector space.

For example, given vectors (v_1), (v_2), and (v_3), a linear combination would look like (c_1 v_1 + c_2 v_2 + c_3 v_3), where (c_1), (c_2), and (c_3) are scalars. This concept is fundamental in linear algebra since it helps describe the relationships between vectors and allows for the solution of linear equations, understanding vector spaces, and many other applications.

The other choices do not define a linear combination correctly. The first choice misrepresents the concept by describing only the product of vectors by scalars rather than their summation. The third option refers to a set of independent vectors, which relates to vector independence rather than combinations. Lastly, the concept of a mixture of vector