Invertible linear maps, linear isomorphisms, and inverse linear maps

Depends on

• Linear map between vector spaces over the same field
• Injection, surjection, bijection

Used by

• Congruent matrices have the same rank; hence rank and nondegeneracy of a bilinear form are basis-independent Corollary
• Determinant is a group homomorphism GL(V)→ F^×, and det(T⁻¹)=det(T)⁻¹ Corollary
• Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension Corollary
• A finite-dimensional representation ρ:G→ GL(V) over a field, and its degree Definition
• Conjugation and the adjoint representation of a Lie group Definition
• Nondegenerate fixed point Definition
• Rational representations and comodules of an affine group scheme Definition
• The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map μ(a) = −2(c−1)⁻¹a Definition
• The canonical reflection homomorphism, roots, reflections, and the positive cone Definition
• The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset Definition
• The geometric representation on the simple-root basis over a common splitting field, and the root set Definition
• A compact group with no faithful continuous finite-dimensional representation Example
• The exponential map of a flat torus is not injective Example
• An isolated fixed point splits under perturbation, preserving its index Lemma
• Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
• Descent of the reflection representation, unit root norms, and conjugation of reflections Lemma
• Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet Lemma
• Graph-diagonal transversality is exactly fixed-point nondegeneracy Lemma
• Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra Lemma
• The adjoint representation of an affine group scheme Lemma
• The general linear group scheme and its coordinate ring Lemma
• The tangent space at the identity is a vector space, and Lie is a functor Lemma
• Conjugate points and multiplicity are invariant under affine reparametrization Proposition
• For a finite-dimensional space, λ is an eigenvalue of T if and only if T-λ I is not invertible Proposition
• Isolated fixed points need not be nondegenerate Remark
• A finitely generated affine group scheme has a faithful finite-dimensional representation Theorem
• A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms Theorem
• Conjugate points are critical values of the exponential map along the geodesic Theorem
• Finiteness criterion: W is finite exactly when the Coxeter form is positive definite Theorem
• Invertible matrix theorem: invertibility, full pivot rank, RREF I, trivial nullspace and unique solvability are equivalent Theorem
• Relative position classifies pairs of complete flags Theorem
• T↦[T]_B^C is a vector-space isomorphism L(V,W)≅ M_m× n(F) Theorem
• The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere Theorem

Dependency tree · two levels

6 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. Schiavone, MIT 18.700 Day 9, Definition 21 (standard reference, not scraped)
• S. Axler, Linear Algebra Done Right, 4th ed., §3D (standard reference, not scraped)