Independent and spanning families

To describe a subspace we look for the smallest set of vectors that suffices to build it. Two qualities pull against each other: having enough (a spanning family) and not having too many (a linearly independent family).

Spanning family: having enough

A family (u1, …, up) spans F if every vector of F can be written as a linear combination of these vectors:

every v in F is written   v = λ1·u1 + λ2·u2 + ... + λp·up

In other words F = Vect(u1, …, up): nothing is missing, the vectors cover the whole target space.

Independent family: not having too many

A family is linearly independent (also called free) if none of its vectors is superfluous. The official criterion:

λ1·u1 + λ2·u2 + ... + λp·up = 0     implies     λ1 = λ2 = ... = λp = 0

Only the trivial combination gives the zero vector. Otherwise the family is dependent: one of the vectors can be expressed from the others and adds nothing.

INDEPENDENT                        DEPENDENT
                                                    u3 = u1 + u2
   ^ u2                              ^ u2          /
   |                                 |            /
   |                                 |  u3 ------/
   +-----> u1                        +-----> u1

two distinct directions           u3 already lies in the plane of u1, u2

The reflexes that save time

- A family containing the zero vector is ALWAYS dependent.
- Two vectors are dependent  <=>  they are collinear (proportional).
- Three vectors in R² are ALWAYS dependent (too many for the dimension).
- More generally: p vectors in a space of dimension n < p are dependent.

That last point is fundamental: you cannot have more independent vectors than the dimension allows.

A worked example

Are the vectors u = (1 ; 2 ; 0), v = (0 ; 1 ; 1) and w = (2 ; 5 ; 1) of R³ independent?

We look for λu + μv + νw = 0. But notice straight away:

2u + v = (2 ; 4 ; 0) + (0 ; 1 ; 1) = (2 ; 5 ; 1) = w

So 2u + v - w = 0 with coefficients not all zero: the family is dependent. The vector w is superfluous, and Vect(u, v, w) = Vect(u, v): three vectors, but only a plane.

When no relation is obvious, the systematic method is to write the vectors as columns and run Gaussian elimination: the family is independent if and only if the rank equals the number of vectors.

The link with systems

These two notions are just a rereading of what we already know about A X = B, where the columns of A are the vectors of the family:

spanning family     <=>  A X = B has AT LEAST one solution, for every B
independent family  <=>  A X = 0 has AT MOST one solution (the zero one)

Summary

• Spanning: every vector of F is a combination of the family — nothing is missing.
• Independent: only the all-zero combination gives 0 — nothing is superfluous.
• Dependent: at least one vector can be expressed from the others.
• A family containing 0, or with more vectors than the dimension, is dependent.
• General method: put the vectors in columns and compute the rank by elimination.