4. Linear Dependence & Independence
In linear algebra, determining whether a set of vectors is linearly dependent or linearly independent is an essential concept.
What is Linear Dependence?
A set of vectors is said to be linearly dependent if at least one vector in the set can be written as a linear combination of the others. This means there is a non-trivial combination (not all scalars zero) that gives the zero vector.
c₁v₁ + c₂v₂ + ... + cₙvₙ = 0
where at least one cᵢ ≠ 0In simple terms: the vectors are “dependent” on each other.
What is Linear Independence?
A set of vectors is linearly independent if the only solution to:
c₁v₁ + c₂v₂ + ... + cₙvₙ = 0
is when c₁ = c₂ = ... = cₙ = 0. This means no vector in the set can be written as a combination of the others.
In simple terms: each vector adds something “new” to the space.
Why is This Important?
- Independent vectors form the foundation (basis) of vector spaces.
- They help us understand dimensions and structures of spaces.
- Used in solving systems of linear equations and in machine learning models.
Examples
Example 1: Dependent Vectors
Let v₁ = [1, 2], v₂ = [2, 4]. Here, v₂ = 2 × v₁.
So, v₁ and v₂ are linearly dependent.
Example 2: Independent Vectors
Let v₁ = [1, 0] and v₂ = [0, 1]. There's no way to write one as a multiple of the other.
So, these vectors are linearly independent.
How to Test Linear Dependence?
You can set up an equation of the form:
c₁v₁ + c₂v₂ + ... + cₙvₙ = 0
Then solve for c₁, c₂, ..., cₙ. If the only solution is all zeros, they’re independent. If any non-zero solution exists, they are dependent.
Visual Insight:
- In 2D: Two vectors are dependent if they lie on the same line.
- In 3D: Three vectors are dependent if they lie in the same plane.