Linear Dependence and Linear Independence

I. Linear Combinations

A linear combination takes several vectors, multiplies each of them by a scalar, and adds the results:

$\mathbf{w} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n$

where $c_1, c_2, \dots, c_n$ are arbitrary scalars (real numbers).

Example: v₁ = (1, 0), v₂ = (0, 1)
w = 3·v₁ + 2·v₂ = (3, 0) + (0, 2) = (3, 2)

Key point: the scalars $c_i$ may be any real number (including negative numbers, zero, and fractions).

II. The Span

Definition

The set of all linear combinations of a group of vectors is called their span:

$\text{span}\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\} = \{c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n \mid c_i \in \mathbb{R}\}$

Intuition: let the scalars run over all real numbers, and you get every position the vectors can “reach”.

Interactive Demo

What to watch:

• Two non-collinear vectors span the whole plane (every point can be reached)
• Two collinear vectors span a single line only
• If the zero vector is included, it contributes nothing, and the span is still determined by the remaining vectors

III. Linear Dependence and Linear Independence

Definition

• Linearly dependent: there exist scalars $c_1, \dots, c_n$, not all zero, such that $c_1\mathbf{v}_1 + \cdots + c_n\mathbf{v}_n = \mathbf{0}$.
• Linearly independent: the linear combination is the zero vector only when $c_1 = c_2 = \cdots = c_n = 0$.

Intuition

Example:
v₁ = (1, 1), v₂ = (2, 2)  → dependent (v₂ = 2v₁)
v₁ = (1, 0), v₂ = (0, 1)  → independent

IV. Bases

Definition

A basis is a set of linearly independent vectors that spans the space, and it does so with nothing to spare:

1. Spanning: linear combinations of the basis vectors reach every point of the space
2. Independence: no vector is redundant (drop any one of them and the span becomes smaller)

The number of vectors in a basis is the dimension of the space. Equivalently, a basis is precisely a maximal linearly independent set: it cannot be enlarged, and the number of vectors in it — the rank of the vector set — is fixed no matter which basis is chosen.

The Standard Basis

The most used basis of the two-dimensional plane is called the standard basis:

$\hat{\imath} = (1, 0), \quad \hat{\jmath} = (0, 1)$

Any vector can be written as a linear combination of them:

$\mathbf{v} = (3, 2) = 3\hat{\imath} + 2\hat{\jmath}$

A Basis Is Not Unique

The same space has infinitely many bases. For instance $\{(1,1), (1,-1)\}$ is also a basis of the two-dimensional plane — as long as the two vectors are not collinear.

Example: check that (1,1) and (1,-1) form a basis
Any (x, y) = a(1,1) + b(1,-1)
Solution: a = (x+y)/2, b = (x-y)/2  ← there is a solution for every x, y
So they span the whole plane, and they are independent → they form a basis

V. An Intuitive Summary

Summary

Symbols Used in This Article