Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Statement
Let and be linear, with ordered bases of , of , and of . Then
Facts & Assumptions
Given: The composable linear maps and ordered bases in the Statement, and a vector .
A composite of linear maps is linear (Identity maps and composites of linear maps are linear).
Matrix multiplication is associative whenever the shapes are compatible (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).
Proof
Apply [L2] to and then to : .
Associativity from [L3] rewrites step 1.1 as for every .
Evaluating at each vector of makes a standard coordinate column, so the columns of equal those of the displayed product. Therefore the matrices are equal.
Depends on
- Identity maps and composites of linear maps are linear
- $[T(v)]_{\mathcal C}=[T]_{\mathcal B}^{\mathcal C}[v]_{\mathcal B}$
- $T\mapsto[T]_{\mathcal B}^{\mathcal C}$ is a vector-space isomorphism $\mathcal L(V,W)\cong M_{m\times n}(F)$
- Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication
Used by
- Congruent matrices have the same rank; hence rank and nondegeneracy of a bilinear form are basis-independent Corollary
- End_F(V) is a ring and matrix representation is a ring isomorphism End_F(V)≅ Mₙ(F) Corollary
- Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
- The quarter-turn (x,y)↦(-y,x) on ℝ² has matrix beginpmatrix0&-11&0 endpmatrix and square -I₂ Example
- In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose Proposition
- [T]_B'^C'=P_C'leftarrowC[T]_B^CP_B leftarrowB' Theorem
- [v]_C=P_C leftarrowB[v]_B and P_B leftarrowC=P_C leftarrowB⁻¹ Theorem
- A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero Theorem
- A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms Theorem
- Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, χ_T(T)=0 Theorem
- For a positive-dimensional finite-dimensional operator, Tadj(T)=adj(T)T=det(T)I Theorem
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then χ_p(T)(y)=∏_i<n(y-p(λᵢ)) for every p∈ F[x]: the eigenvalues of p(T) are p(λᵢ), counted with algebraic multiplicity Theorem
- The (i,j) entry of A(G)ᵏ counts walks of length k Theorem
- The composite of holomorphic maps is holomorphic and its complex Jacobian is the product Theorem
- The minimal polynomial is unchanged by choosing a matrix representation or replacing a matrix by a similar one Theorem
Dependency tree · two levels
13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Definition 3.43 and Theorem 3.81 (standard reference, not scraped)
- S. Schiavone, MIT 18.700 Day 9, Theorem 35 (standard reference, not scraped)