In This Lesson Vectors Matrices 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 Matrix Operations Solving Linear Systems Determinants 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 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‖)
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 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 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 Addition: (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ (same dimensions required) Scalar multiplication: (cA)ᵢⱼ = c·Aᵢⱼ 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 .
Solving Linear Systems 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.
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