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 multiplication: (AB)ᵢⱼ = Σₖ Aᵢₖ·Bₖⱼ (inner dimensions must match)
Inverse: AA⁻¹ = A⁻¹A = I (only for square, non-singular matrices)
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.
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.