knowen.mdknowen.md

Dashboard

Featured nodes

Roots

  • Public root

Templates

  • Guanyu's Latex template
  • Router template

Trees

  • Showcase
Related nodes

Parents1

  • Linear transformations

Siblings39
  • Sort by title
  • Sort by date

  • Eigenvalues and eigenvectors
  • The matrix representation of a sum of linear transformations is the sum of the matrices.
  • A linear transformation is invertible if and only if it is injective and surjective
  • Subspaces associated to a linear transformation
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
  • The preimage of a vector is a translation of the kernel of the linear transformation
  • The matrix representation of a composition of linear transformations is the product of the matrices.
  • 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
  • Terminology
  • Composition
  • 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 image of a spanning set spans the range
  • 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
  • The rank plus the nullity of a linear transformation equals the dimension of the domain.
  • Linear maps preserve dependence of lists, with image-set caveats
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
  • 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 of an inverse is the inverse of the matrix representation
  • Eigenvalues agree and eigenvectors correspond through coordinates
  • Triangularization requires the characteristic polynomial to split
  • Equivalence theorems for injective transformations
knowen.mdknowen.mdBeta
#
  • Explore
  • Sign in
  • Request an invite
  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
526–550 of 718 nodes
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
  • Proof of several equivalences for nonsingular matrix#23644
  • Pythagorean equality characterizes orthogonality over the reals#23918
  • QR decomposition#23630
  • Rank and nullity#23882
  • Rank and nullity#23672
  • Rank–nullity theorem for an m-by-n matrix#23679
  • Real block form and complex diagonalization for a 2-by-2 matrix#23618
  • Removing a linearly dependent vector from a set does not change the span of the set.#23982
  • R^n is a vector space.#24097
  • Row equivalence is an equivalence relation#23480
  • Row-equivalent augmented matrices represent equivalent systems#23769
  • Row equivalent matrices have the same row space.#23881
  • Row operations#23474
  • Row operations are given by multiplication by elementary matrices.#23559
  • Row operations do not necessarily preserve the column space.#23878
  • Scalar multiplication#23470
  • Scalar multiplication of matrices is associative and distributive#23473
  • Similarity of matrices#23634
  • Similarity of matrices is an equivalence relation#23636
  • Similar matrices have the same eigenvalues and the same characteristic polynomials.#23638
« First ← Previous ... 18 19 20 21 22 23 24 25 26 ... Next → Last »
1–4 of 4 public trees
  • Aspen Summer 2016 - Entanglement Matters#1213
  • Container for Linear Algebra#24123
  • Math#23168
  • Test Flight Problem Set#23151

knowen.md© 2026 KTree Inc.

ExploreWhat is knowen.md?PricingIntegrationsContact usLeave feedback