Dashboard

Featured nodes

Roots

  • Public root

Templates

  • iCorps template
  • Guanyu's Latex template
  • Ivar's latex template
  • Family Tree template
  • Router template

Trees

  • Public trees
Related nodes

Parents1

  • Linear transformations

Siblings39
  • Sort by title
  • Sort by date

  • Eigenvalues and eigenvectors
  • Terminology
  • Geometric properties of linear transformations
  • Matrices as linear transformations
  • Basic properties of linear transformations
  • Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
  • Description of a basis for the null space of a matrix from the reduced row-echelon form.
  • The nonzero rows of an echelon form of a matrix are linearly independent.
  • Subspaces associated to a matrix
  • Rank and nullity
  • Examples
  • Composition
  • The preimage of a vector is a translation of the kernel of the linear transformation
  • The image of a linearly independent set under an injective linear transformation is linearly independent.
  • The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
  • The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
  • A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
  • The range of a linear transformation is a subspace
  • The the image of a spanning set is a spanning set for the range space
  • A linear transformation is surjective if and only if the image of a basis is a spanning set
  • Definition of generalized range space of a linear transformation
  • A linear transformation is injective on its generalized range space.
  • Definition of diagonalizable linear transformation
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
  • Subspaces associated to a linear transformation
  • The rank plus the nullity of a linear transformation equals the dimension of the domain.
  • The image of a linearly dependent set under a linear transformation is linearly dependent.
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
  • A linear transformation is invertible if and only if it is injective and surjective
  • Definition of matrix representation of a linear transformation with respect to bases of the spaces
  • A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
  • Definition of matrix representation of a linear transformation from a vector space to itself
  • The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
  • The matrix representation of a sum of linear transformations is the sum of the matrices.
  • The matrix representation of a composition of linear transformations is the product of the matrices.
  • The matrix representation of the inverse of linear transformations is the inverse of the matricix.
  • A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
  • A linear transformation has a representation as an upper triangular matrix.
  • Equivalence theorems for injective transformations
KnowenBeta
#
  • Explore
  • Sign in
  1. A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.

Navigate

Jump to a node in this tree, or explore your trees and public knowledge.

Quick Navigation

Jump to a node or explore another tree.

  • Current tree nodes
  • Public trees
101–125 of 715 nodes
  • Definition of 0 matrix#23458
  • Definition of 0/trivial subspace#23964
  • Definition of 0 vector#23784
  • Definition of adjoint (conjugate transpose)#23506
  • Definition of adjugate/classical adjoint of a matrix#23748
  • Definition of (algebraic) multiplicity of an eigenvalue#23696
  • Definition of a lower triangular matrix#23563
  • Definition of angle between vectors#23915
  • Definition of an upper triangular matrix#23562
  • Definition of applying a polynomial to a linear transformation#24048
  • Definition of applying a polynomial to a square matrix#23700
  • Definition of augmented matrix (of a linear system)#23755
  • Definition of automorphism of a vector space#24007
  • Definition of a vector being orthogonal to a subspace#23919
  • Definition of band matrix#23593
  • Definition of basic/dependent/leading variable in a linear system#23442
  • Definition of basis of a vector space (or subspace)#23829
  • Definition of block diagonal matrix#23570
  • Definition of block/partitioned matrix#23568
  • Definition of change of coordinates matrix between two bases#23995
  • Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)#23832
  • Definition of characteristic equation of a matrix#23702
  • Definition of characteristic polynomial of a linear transformation#24044
  • Definition of characteristic polynomial of a matrix#23701
  • Definition of Cholesky decomposition#23629
« First ← Previous 1 2 3 4 5 6 7 8 9 ... Next → Last »
1–25 of 28 public trees
  • child of test#23311
  • CTSI Resources#23365
  • Electroweak Interaction#1010
  • Field Theories#24290
  • Finance#616
  • How Long Do The Baked Potatoes Take on the Grill#23374
  • Knowen blog#943
  • Lecture Notes#272
  • Lecture notes test#23095
  • Math#23168
  • Media Coverage#953
  • Metrical space#155
  • Misc#929
  • my test article#24378
  • Paid Search#24168
  • Personal Knowledgebases#1202
  • Philosophy#25227
  • Physics#18
  • Products of Knowen#22833
  • Quantum Chromodynamics (QCD)#1009
  • Recyling Bin#23013
  • Security and Strategic Studies#23169
  • TECHNOLOGY and MARKETS#24392
  • Testing#23012
  • test jhjhk#26989
1 2 Next → Last »

Knowen© 2026 KTree Inc.

ExplorePricingIntegrationsContact usLeave feedback