The leap from columns to axioms
Vol I trained you to see a vector as a column in R^n and a vector space as the set of all such columns. That picture is true but small. The real definition throws away the columns and keeps only the behavior: a vector space is a set V together with two operations — addition of vectors and multiplication by scalars — that obey eight rules, the vector space axioms.
Why bother? Because the moment you list only the rules, anything that follows them earns the name 'vector space' — and every theorem you proved in Vol I (about span, basis, dimension) instantly applies. You prove once, reuse everywhere. That is the entire payoff of abstraction.
The eight rules, in plain words
- Closure & a zero: u + v stays in V, and there is a special vector 0 with v + 0 = v. Every vector has a negative -v.
- Addition is tame: it is commutative (u + v = v + u) and associative ((u + v) + w = u + (u + ...) grouped any way).
- Scaling plays nicely: 1*v = v, a*(b*v) = (a*b)*v, and the two distributive laws a*(u + v) = a*u + a*v and (a + b)*v = a*v + b*v.
Three spaces that aren't R^n
The scalars themselves form a field of scalars — usually R or C — supplying the numbers you multiply by. Once that is fixed, here are three genuine vector spaces whose 'vectors' look nothing like arrows.
P_2 = polynomials of degree <= 2: a + b x + c x^2
add coefficient-wise, scale every coefficient. Zero = 0 polynomial.
M_2 = all 2x2 real matrices: [a, b; c, d]
add entry-wise, scale every entry. Zero = [0,0;0,0].
F(R) = all functions f: R -> R
(f + g)(x) = f(x) + g(x), (a*f)(x) = a*f(x). Zero = the function 0.
Check one axiom, e.g. distributivity in P_2:
a*( (1 + x) + (x^2) ) = a*(1 + x + x^2) = a + a x + a x^2
= a*(1 + x) + a*(x^2). OK.The space of all functions, F(R), is the prototype of a function space — and it is huge, far bigger than any R^n. Holding onto it is what eventually forces the infinite-dimensional story in guide 5. For now, just savor that 'vector' has quietly stopped meaning 'arrow'.