Vectors and Linear Combinations

A vector is the first “protagonist” of linear algebra. Its definition takes only two lines — a list of numbers, together with the two operational rules “can be added” and “can be scaled by a scalar” — but from this one step you can grow the concepts of basis, span, coordinates and dimension that run through the whole course.

This chapter first distinguishes the geometric vector (an arrow, a directed segment) from the algebraic vector (an ordered list of numbers), and explains why they are the same thing. It then discusses what properties vector addition and scalar multiplication satisfy, and how they differ in essence from an arbitrary binary operation. Finally it forms the span from all the vectors you can get by “adding and subtracting a few vectors and multiplying them by multiples”, and finds the most economical set of generators of that span — a basis.

By the end of it you should be able to answer: given a set of vectors, how do you decide whether they are linearly independent, and how do you find a basis for the space they span? At that point the seemingly abstract notion of “dimension” becomes something concrete and computable.