Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases

Depends on

• Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
• A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i<n} \lambda_i v_i$ for exactly one $\lambda : n \to F$; those scalars are the coordinates of $x$ in that ordered basis
• Linear map between vector spaces over the same field
• The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Used by

• The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product Corollary
• A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation Definition
• Conjugation and the adjoint representation of a Lie group Definition
• Holomorphic maps ℂᵐ → ℂⁿ and the complex Jacobian matrix Definition
• Jordan blocks, Jordan strings, and their endpoints Definition
• The basis-independent characteristic polynomial χ_T of an endomorphism of a finite-dimensional space, including χ_T=1 in dimension zero Definition
• The basis-independent trace of an endomorphism of a finite-dimensional vector space Definition
• The change-of-basis matrix P_C leftarrowB=[id_V]_B^C Definition
• The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and 1 on the zero space Definition
• The Gershgorin disks of an endomorphism with respect to an ordered basis Definition
• The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space Definition
• Triangularisable endomorphisms and simultaneous triangularisability Definition
• A matrix represents a map F²→ F³ by its images of the standard basis vectors Example
• Changing both domain and codomain bases of a map F²→ F³ uses both sides of the formula Example
• Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
• Componentwise holomorphy checked for an explicit map ℂ²→ℂ³ Example
• The quarter-turn (x,y)↦(-y,x) on ℝ² has matrix beginpmatrix0&-11&0 endpmatrix and square -I₂ Example
• The same operator has two different matrices in two ordered bases Example
• An alternating top-degree form is determined by its value on one ordered basis Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant Lemma
• Complete invariant flags are equivalent to upper-triangular matrices Proposition
• [T(v)]_C=[T]_B^C[v]_B Theorem
• A map into ℂⁿ is holomorphic exactly when each of its components is Theorem
• A vector annihilator gives a power basis and its companion matrix Theorem
• Every eigenvalue lies in some Gershgorin disk Theorem
• Every invertible real or complex square matrix has a unique factorisation A=QR with Q orthogonal or unitary and R upper triangular with positive real diagonal 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
• In basis-wedge coordinates, the matrix of ΛᵏT is the signed matrix of k-minors Theorem
• In dual bases, the matrix of T^* is the transpose of the matrix of T Theorem
• In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix Theorem
• T↦[T]_B^C is a vector-space isomorphism L(V,W)≅ M_m× n(F) Theorem
• The composite of holomorphic maps is holomorphic and its complex Jacobian is the product Theorem
• The geometric multiplicity of an eigenvalue does not exceed its algebraic multiplicity Theorem
• The minimal polynomial is unchanged by choosing a matrix representation or replacing a matrix by a similar one Theorem

Dependency tree · two levels

19 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 3.31 (standard reference, not scraped)