Dashboard

Featured nodes

Roots

  • Public root

Templates

  • Guanyu's Latex template
  • Router template

Trees

  • Public trees
Related nodes

Parents1

  • Subspaces

Siblings16
  • Sort by title
  • Sort by date

  • Definition of subspace
  • Definition of subspace spanned by a set of a set of vectors
  • Definition of the 0/trivial subspace
  • Definition of 0/trivial subspace
  • A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations
  • Definition of intersection of subspaces
  • The intersection of subspaces is a subspace
  • Definition of sum of subspaces
  • The sum of subspaces is a subspace
  • Definition of direct sum of subspaces
  • The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.
  • Definition of independent subspaces
  • A vector can be written uniquely as a linear combination of vectors from independent subspaces.
  • The union of bases from independent subspaces is a basis for the space.
  • Definition of complement of a subspace
  • Theorem characterizing when a space is the direct sum of two subspaces
KnowenBeta
#
  • Explore
  • Sign in
  1. Definition of independent subspaces

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
701–715 of 715 nodes
  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.#23918
  • Unitary matrices#23587
  • Unitary matrices are invertible.#23589
  • Unitary matrices have orthogonal (orthonormal) rows/columns.#23590
  • Unitary matrices preserve inner products.#23591
  • Unitary matrices preserve orthogonal (orthonormal) bases.#23592
  • Using matrices to solve linear systems#23760
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • Vector spaces with the same dimension are isomprphic.#24005
  • Vector sum/addition interpreted geometrically in R^n (or C^n)#23797
  • Vector sum/addition is commutative and associative#23787
  • Visualise a linear transformation on R^2 by looking at the image of a region#23852
« First ← Previous ... 25 26 27 28 29
1–2 of 2 public trees
  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

Knowen© 2026 KTree Inc.

ExplorePricingIntegrationsContact usLeave feedback