Algebra for Notation and Geometry

📘 Matrix & Vector Notation

🔢 Matrices and Vectors

• A, B, C → Matrices (capital letters)
• u, v, w → Vectors (lowercase letters)

🔹 Matrix Size

• A ∈ ℝᵐˣⁿ or A (m × n)
  → Matrix A has m rows and n columns

Example:
If A is 3 × 2, it has 3 rows and 2 columns.

🔹 Transpose

• Aᵀ → Transpose of matrix A
• vᵀ → Transpose of vector v

Transpose flips rows into columns.

🔹 Inverse and Determinant

• A⁻¹ → Inverse of matrix A
  (Only defined for square matrices with non-zero determinant)

• det(A) → Determinant of matrix A

🔹 Multiplication

• AB → Matrix multiplication of A and B
  (Valid only if inner dimensions match)

• u · v or ⟨u, v⟩ → Dot product of vectors

Dot product formula:

$$u \cdot v = \sum_{i=1}^{n} u_i v_i$$

🔹 Sets of Numbers and Vectors

• ℝ → Set of real numbers
  Example: 0, −0.642, 2, 3.456

• ℝ² → Set of 2-dimensional vectors

Example:

$$v = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$$

• ℝⁿ → Set of n-dimensional vectors

• v ∈ ℝ² → Vector v belongs to ℝ²

🔹 Vector Norms

• ‖v‖₁ → L1 norm

$$\|v\|_1 = \sum |v_i|$$

• ‖v‖₂, ‖v‖ → L2 norm (Euclidean norm)

$$\|v\|_2 = \sqrt{\sum v_i^2}$$

🔹 Transformations

• T : ℝ² → ℝ³
  → T maps vectors from 2D space to 3D space

• T(v) = w
  → Vector v ∈ ℝ² is transformed into w ∈ ℝ³