Bases and dimension

A family that is both independent and spanning is called a basis. It is the most useful object in the whole course: it gives every vector coordinates, and every space a number — its dimension.

Definition and fundamental property

BASIS = INDEPENDENT family (nothing superfluous) + SPANNING (nothing missing)

The point lies in this theorem: if (e1, …, en) is a basis of E, then every vector v of E can be written

v = x1·e1 + x2·e2 + ... + xn·en

in exactly one way. The numbers (x1, …, xn) are the coordinates of v in that basis.

Uniqueness comes from independence, existence from spanning. This is what lets us replace an abstract vector (a polynomial, a function, a matrix) by a plain list of numbers, and hence hand it to a computer.

The canonical basis

Each usual space has an "obvious" basis:

R³ :          e1 = (1;0;0)   e2 = (0;1;0)   e3 = (0;0;1)
polynomials
of degree ≤ 3 :  1  ,  X  ,  X²  ,  X³
2×2 matrices  :  the four matrices with a single 1

In R³, writing v = (3 ; -1 ; 2) already gives its coordinates in the canonical basis: 3e1 - e2 + 2e3.

Dimension

All bases of a given space have the same number of elements. That number is the dimension of the space.

Space                           Usual basis               Dimension
------------------------------  ------------------------  ---------
R^n                             e1, ..., en                   n
polynomials of degree ≤ 3       1, X, X², X³                  4
2 × 3 matrices                  the 6 elementary matrices     6
{0}                             (none)                        0
functions from R to R           —                          infinite

Watch out for the classic off-by-one: polynomials of degree ≤ n form a space of dimension n + 1, because of the constant term.

Counting is enough

Here is the result that saves the most work. In a space of dimension n:

n INDEPENDENT vectors    ->  it is a basis (no need to check the rest)
n SPANNING vectors       ->  it is a basis
fewer than n vectors     ->  never spanning
more than n vectors      ->  always dependent

In other words, with the right number of vectors, only one of the two properties needs checking. In practice we check independence, which is usually quicker.

The rank of a family

The rank of a family of vectors is the dimension of the space it spans:

rank(u1, ..., up) = dim Vect(u1, ..., up)

rank = p         ->  the family is independent
rank < p         ->  the family is dependent, p - rank vectors are superfluous

Take u = (1;2;0), v = (0;1;1), w = (2;5;1) again with w = 2u + v: the family has three vectors but rank 2. It spans a plane, not R³.

The incomplete basis theorem

Every independent family can be extended to a basis, and from every spanning family a basis can be extracted. These two results guarantee that we can always get back to the right number of vectors: neither too many nor too few.

Summary

• A basis is an independent and spanning family.
• In a basis, every vector has unique coordinates.
• The dimension is the number of vectors in a basis — the same for all of them.
• dim R^n = n; polynomials of degree ≤ n have dimension n + 1.
• With exactly n vectors in dimension n, independent ⟺ spanning ⟺ basis.
• The rank of a family is the dimension of the space it spans.