Structure

Summary

Structure (Part IV: Expanding) Chapters: 11, 12, 13, 14 Plane Position: (-0.3, 0.5) radius 0.4 Primitives: 51

Linear algebra and higher-dimensional thinking. Vectors, matrices, transformations — the architecture of mathematical space.

Key Concepts: Vector Definition, Vector Space Axioms, Dot Product (Inner Product), Matrix Definition and Operations, Linear Transformation

Key Primitives

Vector Definition (definition): A vector v in R^n is an ordered n-tuple v = (v_1, v_2, ..., v_n) where each v_i is a real number. Vectors represent both magnitude and direction in n-dimensional space.

• I need to represent a quantity with both magnitude and direction
• How do I work with points or directions in multiple dimensions
• Describe a displacement or velocity in n-dimensional space

Vector Space Axioms (axiom): A vector space V over a field F is a set with two operations (addition, scalar multiplication) satisfying 8 axioms: closure under addition and scalar multiplication, commutativity and associativity of addition, existence of zero vector and additive inverses, and distributive laws connecting addition with scalar multiplication.

• Is this set a vector space
• Verify the axioms for a proposed vector space structure
• What properties must a space have to support linear algebra

Dot Product (Inner Product) (definition): The dot product of u, v in R^n is u . v = sum_{i=1}^{n} u_i * v_i. Geometrically, u . v = ||u|| ||v|| cos(theta) where theta is the angle between u and v.

• Find the angle between two vectors
• Compute the projection of one vector onto another
• Check if two vectors are perpendicular

Matrix Definition and Operations (definition): An m x n matrix A is a rectangular array of scalars with m rows and n columns: A = [a_{ij}] where 1 <= i <= m, 1 <= j <= n. Matrix addition is componentwise; scalar multiplication scales all entries.

• Represent a linear system of equations in compact form
• Store and manipulate tabular numerical data
• Encode a linear transformation as a matrix

Linear Transformation (definition): A function T: V -> W between vector spaces is a linear transformation if T(u+v) = T(u)+T(v) and T(cv) = cT(v) for all u,v in V and scalars c. Equivalently, T(c_1v_1 + c_2v_2) = c_1T(v_1) + c_2T(v_2).

• Define a function that preserves vector space structure
• Represent a geometric transformation as a matrix
• Map between different vector spaces while preserving linearity

Eigenvalue and Eigenvector (definition): A scalar lambda is an eigenvalue of a square matrix A if there exists a nonzero vector v such that Av = lambdav. The vector v is the corresponding eigenvector. The set of all eigenvectors for lambda (plus 0) is the eigenspace E_lambda = ker(A - lambdaI).

• Find the directions that a transformation merely scales
• Analyze the long-term behavior of a dynamical system
• Diagonalize a matrix for easier computation

Gradient (definition): The gradient of a scalar field f: R^n -> R is the vector of partial derivatives: grad(f) = nabla f = (df/dx_1, df/dx_2, ..., df/dx_n). It points in the direction of steepest ascent and its magnitude is the rate of maximum increase.

• Find the direction of steepest increase of a function
• Compute the rate of change in an arbitrary direction via directional derivative
• Set up optimization conditions: critical points where nabla f = 0

Analyticity (Holomorphic Function) (definition): A complex function f is analytic (holomorphic) at z_0 if f'(z_0) = lim_{h->0} (f(z_0+h)-f(z_0))/h exists, where h approaches 0 through complex values. f is entire if analytic on all of C. Analytic implies infinitely differentiable and equals its Taylor series.

• Determine if a complex function is differentiable in the complex sense
• Identify functions that have power series representations
• Apply the powerful theorems of complex analysis to a function

Vector Addition and Scalar Multiplication (definition): For vectors u, v in R^n and scalar c in R: (u+v)_i = u_i + v_i (componentwise addition) and (cv)_i = c * v_i (scalar multiplication). These operations satisfy closure, commutativity, associativity, and distributivity.

• How do I add or scale vectors
• Combine forces or velocities acting on an object
• Compute a linear combination of vectors

Vector Norm (Magnitude) (definition): The Euclidean norm of v in R^n is ||v|| = sqrt(v . v) = sqrt(sum_{i=1}^{n} v_i^2). It measures the length (magnitude) of the vector. A unit vector has ||v|| = 1.

• Find the length or magnitude of a vector
• Normalize a vector to unit length
• Compute distance between two points in R^n

Composition Patterns

• Vector Definition + structure-vector-addition -> Vector arithmetic in R^n (sequential)
• Vector Addition and Scalar Multiplication + structure-vector-definition -> Complete vector arithmetic system (parallel)
• Vector Space Axioms + structure-linear-transformation -> Theory of linear maps between vector spaces (sequential)
• Dot Product (Inner Product) + structure-vector-norm -> Angle measurement between vectors: cos(theta) = (u.v)/(||u|| ||v||) (sequential)
• Cross Product + structure-dot-product -> Scalar triple product: u . (v x w) = volume of parallelepiped (sequential)
• Vector Norm (Magnitude) + structure-vector-definition -> Distance metric in R^n: d(u,v) = ||u-v|| (sequential)
• Orthogonality and Projection + structure-span-basis -> Gram-Schmidt orthogonalization: convert any basis to orthogonal basis (sequential)
• Linear Independence + structure-span-basis -> Dimension of a vector space: size of any basis (sequential)
• Span and Basis + structure-linear-transformation -> Matrix representation of a linear map with respect to chosen bases (sequential)
• Triangle Inequality for Vectors + structure-vector-norm -> Metric space structure on R^n with triangle inequality as key axiom (parallel)

Cross-Domain Links

• perception: Compatible domain for composition and cross-referencing
• change: Compatible domain for composition and cross-referencing
• reality: Compatible domain for composition and cross-referencing
• foundations: Compatible domain for composition and cross-referencing
• mapping: Compatible domain for composition and cross-referencing
• synthesis: Compatible domain for composition and cross-referencing

Activation Patterns

• vector
• matrix
• linear
• space
• dimension
• basis
• eigenvalue
• transformation
• determinant