Lecture Notes: Linear Independence in Vector Spaces

Introduction to Linear Independence

• Goal: Express vector spaces in the simplest, most efficient way with the fewest elements needed.
• Focus on removing unnecessary information.

Basic Example: Vector Dependence

• Vectors: a, b, c (c = a + 2b)
  • Example: Linear combination 2a + b - c
  • c can be replaced by (a + 2b), indicating c is unnecessary.
• Linearly Dependent: c can be written in terms of a and b.
• Linearly Independent: No vector can be expressed in terms of others.

Mathematical Expression

• Vectors v1, v2,..., vn in vector space V
• Equation: c1v1 + c2v2 + ... + cnvn = 0
  • Linearly independent if all scalars (c1 to cn) are zero.
  • Linearly dependent if non-zero solutions exist.

Solving for Independence

• Solving a system of equations to determine dependence or independence.
• Example: Vectors (1, 1) and (1, -1)
  • Set linear combination equal to zero.
  • Solve using substitution or elimination: Only zero solutions imply linear independence.

Complex Systems and Free Variables

• Use matrix and row reduction to solve more complex systems.
• Row Echelon Form: Simplifies determining linear independence.
• Free Variable: Indicates linear dependence when the system has fewer equations than unknowns.

Example with Matrix

• Vectors: (1, 1, 3, 3), (-1, 1, -1, -3), (2, 5, 9, 6)
  • Use row operations to achieve row echelon form.
  • Presence of free variable indicates linear dependence.

Determinant Method for Square Matrices

• Applicable when vectors form a square matrix.
• Determinant = 0: Linearly dependent.
• Determinant ≠ 0: Linearly independent.
• Example: Vectors (1, 1) and (1, -1) have non-zero determinant, confirming independence.

Polynomials and Linear Independence

• Vector spaces include elements beyond vectors (e.g., polynomials).
• Example: Polynomials x^2 + x - 2, x^2 - 3x + 5, 2x^2 + 6x - 11
  • Create matrix and determine independence via determinant.
  • Determinant zero implies dependence.

Conclusion

• Vector spaces unify functions, vectors, matrices, etc., under a common set of tools.
• Key Question: Can objects be expressed in terms of others?
• Next Topic: Combine linear independence with the concept of span.
• Comprehension Check.