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Vectors & Matrices

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The fundamental building blocks of linear algebra and modern computation.

Vectors

A vector is an ordered list of numbers representing magnitude and direction. In ℝⁿ, vectors can represent points, displacements, or abstract data.

Vector: v = (v₁, v₂, …, vₙ)
Magnitude: ‖v‖ = √(v₁² + v₂² + … + vₙ²)
Dot product: u·v = u₁v₁ + u₂v₂ + … + uₙvₙ
cos θ = (u·v)/(‖u‖·‖v‖)

The magnitude formula generalizes the distance formula from coordinate geometry. The dot product connects to trigonometric functions via the angle between vectors.

Matrices

A matrix is a rectangular array of numbers. An m×n matrix has m rows and n columns.

Identity matrix: I (1s on diagonal, 0s elsewhere)
Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ
Symmetric: A = Aᵀ

Special matrices: diagonal, upper/lower triangular, symmetric, orthogonal (QᵀQ = I). The identity matrix I acts like 1 in matrix multiplication — similar to how 1 is the multiplicative identity.

Matrix Operations

  1. Addition: (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ (same dimensions required)
  2. Scalar multiplication: (cA)ᵢⱼ = c·Aᵢⱼ
  3. Matrix multiplication: (AB)ᵢⱼ = Σₖ Aᵢₖ·Bₖⱼ (inner dimensions must match)
  4. Jp+9l8p92In0a8g6FWBDBOx2MykG6fJh07jKqGxDfrvibp6KDwN5deb3cgZO3rZSUtfSk3JElsFsRA+VXseUI5qmPu/liyDw6xE1mhiTrePDAZY+GRFRRgzZFd6cpZZlOXwxdE+lQVMkvBc2NTzkXeJFetpH64aFqd73n5b0Tm4eZtOvc5zxq7mzdpJwfompO4KmdU3FqA0hapI9GSzkzDZKGFDtYNJ+ki61nWQKDvGbT3pfN9qqINv9dFGW1hRjmqY+7+WUJcKr3Ql51UBqOseUycQHYhlhE5/6TdMtgb1x2EE04o3urtdowXqcEsEm9P8zOL/EpsccGK4xpy5CWG7+PrXwQrA5WNfs+tQYQDLf4QGcKz4nSJ/4CqBAEk8b3PuwRrPcwjMoxYEl5nGZTELAZnxNLQMRbBQw1IjiKzgweLtmgnoP3L/3SCI+Hy/mvn4d+H4BcX3v1nSXBp7agIsKq82bjdm6v8QUPEK/wH1RBGkFA4ZMRq85MIRheU2DI1rO9oN6Ac1QWSeT+Tpv3SJiZpJFNCeJ0BVXkgytOj48PkZOXjlxgFrxNrsZkLzJ8tpFkv2lNeIGOoGJiL3YXZb/MXCe6WcWnvJZxqPoCkfw4DxikX7FPWjs5LFv67R9GnzlfMJRkWDS9bpLZC0EydCxX5GlRIKBFxas+B8qgmwGsbU18EqCGWVeZnJc0LjBmTdqEmhaZIvPOIqgYZFX1Ad/I9Xz3MITLc3nmbcsxg3AAcazW81IWPfX7
  5. Inverse: AA⁻¹ = A⁻¹A = I (only for square, non-singular matrices)
  6. 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
Matrix multiplication is not commutative: AB ≠ BA in general. However, it is associative: (AB)C = A(BC). This algebraic structure is fundamental to linear transformations.

Solving Linear Systems

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Linear systems Ax = b can be solved via:

  • Gaussian elimination: Row reduce to echelon form
  • Matrix inverse: x = A⁻¹b (when A is invertible)
  • Cramer's rule: xᵢ = det(Aᵢ)/det(A)
  • 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

These generalize the methods for solving systems of equations to any number of variables. In regression analysis, the normal equations have the form b = (XᵀX)⁻¹Xᵀy.

Determinants

2×2: det(A) = ad − bc
3×3: cofactor expansion along any row/column
Properties: det(AB) = det(A)·det(B), det(Aᵀ) = det(A)

The determinant encodes whether a matrix is invertible (det ≠ 0), the scaling factor of the associated transformation, and the signed volume of the parallelepiped formed by column vectors.