Vector space

After working with vectors in \(\mathbb{R}^n\) one notices that many other objects are added and scaled in the same way: matrices, polynomials, continuous functions. Axler and Strang proceed like this: list the properties that are really used and call vector space any set satisfying them. Whatever is proved from those axioms holds at once for every example.

A guiding example: polynomials of degree \(\le 2\) form a space, with the polynomial \(0\) as zero vector and \(-p\) as the inverse of \(p\). Polynomials of degree exactly 2 do not, because \((t^2+t)+(-t^2+1)=t+1\) leaves the set. Another odd example: on \(\mathbb{R}_{>0}\) with \(u\oplus v=uv\) and \(c\odot u=u^c\), the zero vector is the number \(1\).

Typical mistakes: thinking the zero vector is always \((0,\dots,0)\) (it depends on the space), checking only a couple of axioms, or confusing the scalar \(0\) with the zero vector. Immediate consequences such as \(0v=0\) and \((-1)v=-v\) follow from the axioms.

It connects with what came before because \(\mathbb{R}^n\) is the first example, and with what follows: a vector subspace asks when a subset of a vector space is, with the same operations, a space by itself. The eight axioms are: commutativity and associativity of addition, existence of the zero vector and of the additive inverse \(-v\), \(1v=v\), \(c(dv)=(cd)v\) and both distributive laws.