Linear Algebra

Phase 1: The Foundations

Syllabus Goal

Master linear vector spaces, matrices, linear operators, and eigenvalues/eigenvectors.

Core Concepts

• Vector Space: A set of vectors closed under addition and scalar multiplication.
• Basis: A linearly independent spanning set.
• Inner Product: $\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u}^\dagger \mathbf{v}$.

Matrices and Operators

$$ A \mathbf{x} = \lambda \mathbf{x} $$ where $\lambda$ is the eigenvalue and $\mathbf{x}$ is the eigenvector.

• Trace: Sum of diagonal elements.
• Determinant: Volume scaling factor.
• Hermitian Matrix: $A^\dagger = A$ (Has real eigenvalues).
• Unitary Matrix: $U^\dagger = U^{-1}$ (Preserves inner products).

Implementation in Zig

• Representing vectors and matrices with arrays.
• Implementing Gram-Schmidt orthogonalization.
• Eigenvalue algorithms (e.g., Jacobi method).