Linear Independence, Basis and Dimension Explained

A basis: enough vectors, and not one too many.

Vectors are independent when none of them is a combination of the others. A basis is an independent set that reaches the whole space. Every basis of a space has the same size: its dimension.

independent + spans the space = basis size of every basis = dimension

Think: a minimal toolkit

A good toolkit lets you do every job, and has no tool that another tool already covers. Every job doable means the vectors span the space. No redundant tool means they are independent. Both together make a basis.

The dimension is how many tools that takes. On a map, every place needs exactly two numbers. For a drone in the air, three. That count does not depend on which directions you pick.

Tests for independence

For the columns of A, these all say the same thing:

The only combination giving zero is the all-zero one: Ax = 0 only for x = 0.
N(A) is just the zero vector.
Every column has a pivot: rank r = n.

Four vectors in 3D are always dependent. There is no room for a fourth independent direction.

Worked example

v₁ = (1, 0, 1), v₂ = (0, 1, 1), v₃ = (1, 1, 2).

v₃ = v₁ + v₂, so 1·v₁ + 1·v₂ − 1·v₃ = 0

Dependent. The rank is 2. All three lie in one plane, and v₁, v₂ alone are a basis for it. The plane has dimension 2.

Bases are not unique

(1, 0) and (0, 1) are a basis for the plane. So are (1, 1) and (1, −1). Different tools, same number of them. The rank of A is the dimension of its column space.