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 faithful canonical realization that is not reflection faithful: the affine rank-two system Counterexample
• 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
• Dirac Notation Conversion Definition
• Finite POVM outcome kernel Definition
• Holomorphic maps ℂᵐ → ℂⁿ and the complex Jacobian matrix Definition
• Jordan blocks, Jordan strings, and their endpoints Definition
• Partial trace over a finite tensor factor 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
• Fredholm determinant of a finite-rank operator Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
• The finite dihedral rotation and the infinite unipotent rank-two product 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
• A finite instrument with nonzero input has nonzero output space Lemma
• A number-field unit is exactly an algebraic integer of norm plus or minus one Lemma
• A positive Choi matrix yields finite Kraus operators Lemma
• An alternating top-degree form is determined by its value on one ordered basis Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• Local separable trace-class determinant construction Lemma
• Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order Lemma
• Tensor products of operator matrix units form a basis Lemma
• The dual action, the faces, and the rank-two chamber tiling Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness Lemma
• The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant Lemma
• The trace pairing on a finite endomorphism space is nondegenerate Lemma
• Complete invariant flags are equivalent to upper-triangular matrices Proposition
• Trace Holder Inequality Proposition

…and 23 more results.

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)