Kernel and image of a linear map

Depends on

• Linear map between vector spaces over the same field

Used by

• The dimension of the space of level-one modular forms Corollary
• The sharp real phase-retrieval count for full-spark measurements Corollary
• Admissible constraint classes and restricted identifiability Definition
• Centered conditional modulus of continuity Definition
• Complement splitting and range projection for a simple fold Definition
• Density operator Definition
• Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism Definition
• Finite-dimensional quantum relative entropy Definition
• Quantum linear-systems state-output task Definition
• Rank and nullity of a linear map with finite-dimensional domain Definition
• Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator Definition
• The dual action, chambers, faces, and root hyperplanes Definition
• The tangent space to a regular level set Definition
• A null normal admits no reflection of the displayed form Example
• Conjugate antipodes on the round sphere Example
• The quotient by the kernel is isometric to the range with its induced quotient norm Example
• Two-state reversible walk and its endpoint phases Example
• A linear functional annihilating the kernel of a surjection is a unique transpose multiple Lemma
• Disconnected diagrams, direct products, and comparison of invariant forms Lemma
• Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet Lemma
• Extending a basis of the kernel to a basis of the domain gives a basis of the image Lemma
• Functionals vanishing on a common kernel are combinations of an independent family Lemma
• Graph charts and tangent directions for complex rank-r projectors Lemma
• Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary Lemma
• Kernel and rank sequences of powers stabilise once equality occurs Lemma
• Kernels, images, quotients, and matrix coefficients of comodules Lemma
• Moved space of a reversed reflection product with independent normals Lemma
• Orthogonal complement of an eigenspace is invariant Lemma
• Positive square root of a compact positive operator Lemma
• Rank bookkeeping for a long exact sequence of finite-dimensional vector spaces Lemma
• Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order Lemma
• Singular values equal approximation numbers Lemma
• Tensoring injections and the kernel of a tensor product of quotient maps over a field Lemma
• The dual action, the faces, and the rank-two chamber tiling Lemma
• The nilpotent Lefschetz filtration is canonical and self-dual Lemma
• The QLSP target is nonzero Lemma
• A constraint removes kernel ambiguity exactly when its differences avoid the kernel Proposition
• A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not Theorem
• Cohomology of O(d) on projective space Theorem
• Conditional state for a projective measurement Theorem

…and 9 more results.

Dependency tree · two levels

3 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

• Axler, Linear Algebra Done Right, Chapter 3 (standard reference, not scraped)