Understanding_Linear_Dependence.md

Definition

(from: https://en.wikipedia.org/wiki/Linear_independence)

A sequence of vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ from a vector space ${{mvar|V}}$ is said to be 'linearly dependent', if there exist scalars $a_1, a_2, \dots, a_k,$ not all zero, such that : $a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \cdots + a_k\mathbf{v}_k = \mathbf{0},$ where $\mathbf{0}$ denotes the zero vector.

Understanding Linear Dependence

The statement defines linear dependence of vectors. Let's break down why this definition makes sense:

Core Idea: A set of vectors is linearly dependent if you can create one of the vectors as a linear combination of the others. In other words, one vector becomes redundant because it doesn't add any new information to the set.

Equation Breakdown:

• α₁v₁ + α₂v₂ + ... + αₖvₖ = 0: This equation represents a linear combination of the vectors v₁, v₂, ..., vₖ, where α₁, α₂, ..., αₖ are scalars (coefficients).
• Not all α's are zero: This is crucial. If all α's were zero, the equation would trivially hold for any set of vectors, making the concept of linear dependence meaningless.
• Equals zero vector: This implies that the linear combination of the vectors results in the zero vector, meaning they effectively cancel each other out:
  • Cancellation: Each vector in the combination contributes some direction and magnitude to the sum. When the sum equals the zero vector, it implies that these contributions effectively cancel each other out. In other words, the vectors collectively "undo" each other's effects.
  • Independence Check: When exploring linear dependence, finding that the linear combination equals the zero vector is a critical test. If this is the case, it suggests that the vectors are related in such a way that their combined influence is nullified.
  • Criterion for Dependence: If the only way to form the zero vector from a set of vectors is by creating a non-trivial linear combination (i.e., not all coefficients being zero), then these vectors are linearly dependent. This means that at least one vector in the set can be expressed as a linear combination of the others. In essence, it is redundant because its influence is captured by the combination of the remaining vectors.

Why is this true?

1. Redundant Vector: If the equation holds with at least one non-zero α, we can rearrange it to express one vector in terms of the others. For instance, if α₁ is not zero, we can isolate v₁:

v₁ = (-α₂/α₁)v₂ - ... - (αₖ/α₁)vₖ

This shows that v₁ is a linear combination of the other vectors, making it redundant in the set.

2. Geometric Interpretation: Imagine the vectors as arrows in space. Linear dependence means these arrows can be combined (stretched and added) to form a closed loop, ending up back at the origin (zero vector).

3. Dimension and Span: Linearly dependent vectors do not contribute to the "dimension" of the space they span. They lie within the same subspace as some combination of the other vectors.

Examples:

• Two collinear vectors: If two vectors lie on the same line, one is a scalar multiple of the other, making them linearly dependent.
• Three coplanar vectors: If three vectors lie on the same plane, one can be expressed as a linear combination of the other two.

Opposite - Linear Independence:

If the equation only holds true when all α's are zero, the vectors are linearly independent, meaning none of them can be expressed as a linear combination of the others. They contribute unique information to the space they span.

Importance:

The concept of linear dependence is crucial in linear algebra. It helps us understand vector spaces, bases, dimension, and the solvability of linear systems.