linear combination

A linear combination is the most basic move in all of linear algebra: take a few vectors, stretch or shrink each one by some number (a scalar), and add the results. That is it. If v and w are vectors, then 3*v + 2*w is a linear combination of them: you scaled v by 3, scaled w by 2, and added. The numbers are called coefficients or weights, and they may be any real numbers, including zero and negatives.

This simple operation is the engine behind almost every idea you will meet later. Asking 'which vectors can I build out of these?' leads to the notion of a span. Asking 'is one of these vectors redundant because it is already a combination of the others?' leads to linear independence. Asking 'how do I write a target vector as a combination of a chosen set?' is exactly solving a system of linear equations.

A tiny example: with v = (1, 0) and w = (0, 1), the combination 4*v + 3*w = (4, 3). By choosing different weights you can reach every point in the plane, which is what makes these two vectors special. With a single vector you could only reach points along one line through the origin.