Spaces and subspaces

What is a vector space?

A vector space is a set whose elements can be added together and multiplied by a number without ever leaving the set. This deliberately sparse definition is where its power lies: once it has been verified somewhere, every theorem of linear algebra applies there.

Two operations, one requirement

In a vector space E we have:

addition             u + v      (two vectors -> a vector)
scalar               λ · u      (a number and a vector -> a vector)
   multiplication

The only thing that really matters fits in one sentence: every combination λu + μv stays in E. We say E is closed under linear combination. It follows immediately that E contains the zero vector 0 (take λ = μ = 0) and the opposite of each of its elements.

The official list has eight axioms (associativity, commutativity, distributivity, neutral element, opposites…), but in practice they never cause trouble: they are the usual rules of arithmetic. Closure is what gets checked.

The model: vectors in the plane

                 v
                 ^
                /|
               / |          u + v
              /  |         /
             /   |       /
            O----+----->/-------> u

  u + v : parallelogram rule
  2u    : same direction, twice the length
  -u    : same direction, opposite way

R² (pairs of numbers) and R³ (triples) are the reference examples. More generally R^n, the set of lists of n numbers, is a vector space.

The real point: "vectors" that do not look like any

Here is why the definition is built this way. These sets are all vector spaces:

Set                               A "vector" there is...
--------------------------------  ---------------------------------
R^n                               a list of n numbers
polynomials of degree ≤ 3         3x² - x + 5
functions from R to R             sin, exp, x -> x² + 1
2 × 3 matrices                    a table of 6 numbers
real sequences                    (1 ; 1/2 ; 1/4 ; 1/8 ; ...)
solutions of y'' + y = 0          a function, here a·cos + b·sin

Check it on polynomials: the sum of two polynomials of degree at most 3 still has degree at most 3, and so does the product by a number. It is closed, so it is a vector space. We can then speak of the dimension of that set, of a basis, of the rank of a map — the whole vocabulary becomes available.

This is the typical move in mathematics: isolate the minimal property (here, closure), and everything proved from it holds at once in dozens of different settings.

A counterexample to fix ideas

The set of polynomials of degree exactly 2 is not a vector space:

(x² + x) + (-x² + 1) = x + 1        degree 1, outside the set!

Likewise, the set of vectors of R² with positive coordinates is not one either: multiplying by -1 takes you out of it.

Summary