Linear Dependence

Definition

A set of vectors ${v_1, v_2, \dots, v_n}$ from a Vector Space $\mathbb{V}$ is linearly dependent if there exist scalars $c_1, c_2, \dots, c_n$, not all zero, such that:

If no such scalars exist (or equivalently, if $c_1v_1 + c_2v_2 + \dots + c_nv_n = \mathbf{0}$ implies all $c_i = 0$), then the vectors are linearly independent.

Visual Examples

Examples

In $\mathbb{R}^2$

Consider the vectors:

• $v_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix}$
• $v_2 = \begin{pmatrix} 2 \ 0 \end{pmatrix}$

These are linearly dependent because $2v_1 - v_2 = \mathbf{0}$

In Polynomial Space

The polynomials $p_1(x) = x^2 + x$ and $p_2(x) = x^2 + x - 1$ are linearly independent because no non-zero scalar multiple of one equals the other.

Properties

1. Any set of vectors containing $\mathbf{0}$ is linearly dependent
2. Any set containing a vector that's a Linear Combination of others is dependent
3. A single non-zero vector is always linearly independent
4. Adding a vector to a linearly independent set might create dependence

Why Do We Care?

Linear independence helps us:

1. Find Bases for vector spaces
2. Determine minimal generating sets
3. Calculate Dimensions of spaces
4. Solve systems of equations efficiently

Think of linear independence as a way to identify when vectors are truly "different" from each other - when none can be created from combinations of the others.

Exercise

Prove that if ${v_1, v_2, \dots, v_n}$ is linearly dependent, then one of the vectors can be written as a linear combination of the others.