Vector Operations

There are only three operations on vectors — addition, subtraction and scalar multiplication — and yet they satisfy a whole set of axioms, which is precisely where the word “linear” comes from.

The operational laws of addition and scalar multiplication look unremarkable, and yet they play a key role over and over again later on: whether a set of vectors is linearly dependent, whether it can serve as a basis, and why matrix multiplication can be defined the way it is, all rest on these few axioms. The section on linear combinations is especially important — it is the hub from which every later concept hangs.

This chapter also explains the geometric meaning of these operations: vector addition corresponds to the parallelogram rule, and scalar multiplication to scaling. The two explanations, algebraic and geometric, have to be translatable into one another at any moment.