Foundations

Scientific relationships · 23 / 23

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

1. Linear maps and matrix representation → Vectors
   Uses definition
   Relationship sources · 1

   • Relationship source ↗

   Linked statements

   • A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
2. Linear maps and matrix representation → Functions, domain and codomain
   Uses definition
   Relationship sources · 1

   • Relationship source ↗

   Linked statements

   • A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
3. Rank–nullity theorem → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Linear Algebra 1: Introduction ↗

   Linked statements

   • For a linear map T:V→W with finite-dimensional V, dim V = dim ker T + dim im T. Rank is the image dimension and nullity is the kernel dimension.
4. Real singular value decomposition → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Linear Algebra, Lecture 29: Singular value decomposition ↗

   Linked statements

   • Every real m×n matrix A admits A=UΣVᵀ with orthogonal U,V and rectangular diagonal Σ having nonnegative entries. These singular values are square roots of eigenvalues of AᵀA, with zero padding as needed.
5. Total derivative in finite dimensions → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Lecture 3, Definition 2.3 ↗

   Linked statements

   • For f:U⊂Rⁿ→Rᵐ with U open, f is differentiable at a if a linear map B satisfies ‖f(a+h)−f(a)−Bh‖/‖h‖→0 as h→0. This B is the total derivative Df(a).
6. Constant-coefficient linear ODE systems → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Ordinary Differential Equations, chapter 3 ↗

   Linked statements

   • For a constant real n×n matrix A, x′=Ax with x(0)=x₀ has the unique global solution x(t)=eᵗᴬx₀, where eᵗᴬ=Σₖ₌₀∞ tᵏAᵏ/k! converges for every real t.
7. LU factorization and triangular solves → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Fundamentals of Numerical Computation: LU factorization ↗

   Linked statements

   • An LU factorization writes A=LU with unit lower-triangular L and upper-triangular U. When U has nonzero diagonal, solving Ly=b then Ux=y solves Ax=b.
8. Orthogonal matrices and QR factorization → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Fundamentals of Numerical Computation: The QR factorization ↗

   Linked statements

   • A real orthogonal square matrix Q satisfies QᵀQ=I and preserves Euclidean lengths. For full-column-rank A∈Rᵐˣⁿ, m≥n, a thin QR factorization A=QR has orthonormal Q columns and nonsingular upper-triangular R.
9. Condition number and linear-system perturbation → Linear maps and matrix representation
   Uses definition
   Relationship sources · 1

   • Fundamentals of Numerical Computation: Conditioning of linear systems ↗

   Linked statements

   • For invertible A and an induced matrix norm, κ(A)=‖A‖‖A⁻¹‖. If A(x+Δx)=b+Δb with b≠0, then ‖Δx‖/‖x‖≤κ(A)‖Δb‖/‖b‖.
10. Gaussian elimination and consistency → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Interactive Linear Algebra, section 1.2: Row reduction ↗

    Linked statements

    • Swapping rows, scaling a row by a nonzero scalar and adding a multiple of another row preserve a linear system's solution set. Reduction to echelon form followed by back substitution identifies pivots, free variables and inconsistent equations 0=c≠0.
11. Determinants and invertibility → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Notes on Diffy Qs: Determinant ↗

    Linked statements

    • With Aᵢⱼ formed by deleting row i and column j, cofactor expansion gives det A=Σⱼ(−1)ⁱ⁺ʲaᵢⱼdet Aᵢⱼ, starting from det[a]=a. It satisfies det(AB)=det A det B; A is invertible exactly when det A≠0.
12. Complementary and particular solutions → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Notes on Diffy Qs: Nonhomogeneous equations ↗

    Linked statements

    • For a linear equation Ly=f, any one particular solution yₚ plus the general homogeneous solution y_c gives every solution y=y_c+yₚ. Undetermined coefficients tries polynomial/exponential/trigonometric forcing forms, multiplying by sufficient powers of t when the trial overlaps a homogeneous mode.
13. Time-homogeneous Markov chains → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Markov Chains, section 1.2: Definitions ↗

    Linked statements

    • A chain satisfies P(Xₙ₊₁=j|X₀,…,Xₙ)=Pᵢⱼ when Xₙ=i, where P has nonnegative entries and each row sums to one. With row-vector initial law μ, the law after n steps is μPⁿ.
14. Systematic binary linear code construction → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Abstract Algebra: Theory and Applications: Parity-check and generator matrices ↗

    Linked statements

    • Over F₂ take H=[A|Iₘ] and G=[Iₙ₋ₘ;A]. Encoding message x as c=Gx is injective, HG=0, and the code is exactly ker H. The first n−m coordinates hold information and the final m supply parity checks.
15. Basis coordinates and similarity → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Interactive Linear Algebra, section 4.3: Change of basis ↗

    Linked statements

    • If columns of invertible P are an ordered basis B in standard coordinates, v=P[v]B. A linear operator with standard matrix A has basis matrix [T]B=P⁻¹AP. A new matrix representation changes coordinates, not the underlying map.
16. Affine transformations and invariants → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Math 133: Transformational geometry, isometries and affine transformations ↗

    Linked statements

    • An invertible affine transformation of ℝⁿ has form T(x)=Ax+b with A invertible. It preserves affine combinations whose weights sum to one, collinearity, parallelism and the signed parameter along a line. General affine transformations need not preserve Euclidean lengths or angles.
17. Discrete Fourier transform and inverse → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • 6.046J (Spring 2012), Lecture 5: Fast Fourier transform ↗

    Linked statements

    • For N complex coefficients aⱼ define yₖ=Σ_(j=0)^(N−1)aⱼωᴶᴷ, with ω=e²πⁱ⁄ᴺ. The inverse is aⱼ=N⁻¹Σ_(k=0)^(N−1)yₖω⁻ᴶᴷ. This converts polynomial coefficients to values at the Nth roots of unity.
18. Cayley–Hamilton matrix identity → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Math220, section 6.A: Cayley–Hamilton theorem ↗

    Linked statements

    • For a finite square matrix A over a field, its characteristic polynomial p(t)=det(tI−A)=tⁿ+cₙ₋₁tⁿ⁻¹+⋯+c₀ satisfies p(A)=Aⁿ+cₙ₋₁Aⁿ⁻¹+⋯+c₀I=0. Polynomial division by p reduces any polynomial in A to degree below n; when c₀≠0 the identity also expresses A⁻¹ as a polynomial in A.
19. Projecting a planar bed into a geological section → Linear maps and matrix representation
    Related to
    Relationship sources · 1

    • §12.4, strike/dip definitions and map-to-section figures12.4.1–12.4.4 ↗

    Linked statements

    • For a locally planar bed with true dip δ, choose horizontal x toward its down-dip direction and elevation z upward. Its plane satisfies z=z₀−x tanδ. A vertical section making horizontal angle β to strike has apparent dip magnitude α with tanα=tanδ|sinβ|. A section normal to strike displays the true dip; a section along strike displays no dip for a nonvertical bed.

    Projection uses a coordinate representation of a plane.

20. Ternary compositions and tie-triangle accounting → Linear maps and matrix representation
    Uses definition
    Relationship sources · 1

    • Ternary phase diagrams: normalized components, liquidus projection and sequential lever balances ↗

    Linked statements

    • Three-component compositions occupy a normalized triangle: x_A+x_B+x_C=1 with nonnegative fractions. In a three-phase equilibrium, the bulk composition is x₀=f₁x₁+f₂x₂+f₃x₃ with f₁+f₂+f₃=1. Nonnegative phase fractions express the bulk point as barycentric coordinates of the tie triangle. Two-phase mixtures lie on their tie line.

    Independent component equations determine the phase fractions.

21. Bounded operators and continuous duals → Linear maps and matrix representation
    Related to
    Relationship sources · 1

    • 18.102 (Spring 2021): Bounded linear operators ↗

    Linked statements

    • A linear map T:X→Y between normed spaces is continuous exactly when ||Tx|| ≤ C||x|| for some finite C. Its operator norm is sup over unit vectors of ||Tx||. The continuous dual X* contains bounded linear functionals; arbitrary algebraic functionals need not be continuous in infinite dimensions. If Y is Banach then bounded operators X→Y form a Banach space in this norm.

    Finite-dimensional linear maps are the bounded special case.

22. Cotangent vectors, tensors and pullback → Linear maps and matrix representation
    Related to
    Relationship sources · 1

    • 18.155 Differential Analysis (2 October 2026) ↗

    Linked statements

    • A covector is a linear functional on the tangent space, and a tensor is multilinear on tangent/cotangent inputs. For a smooth map F:M→N the tangent map dF pushes vectors forward at a point, while covectors and differential forms pull back by composition with dF. The coordinate transformation rule is essential; a table of components alone is not an invariant tensor.

    The derivative is a pointwise linear map.

23. Linear representations and Maschke decomposition → Linear maps and matrix representation
    Related to
    Relationship sources · 1

    • Group Theory: representations of finite groups ↗

    Linked statements

    • A representation of a finite group G is a homomorphism G→GL(V). If the field characteristic does not divide |G|, every invariant subspace in a finite-dimensional representation has an invariant complement: average an arbitrary projection over the group. In characteristic p dividing |G| this averaging denominator is unavailable and complete reducibility can fail.

    Group elements act by invertible linear maps.