Dashboard

Featured nodes

Roots

  • Public root

Templates

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

Trees

  • Public trees
Related nodes

Parents1

  • Linear (in)dependence

Siblings17
  • Sort by title
  • Sort by date

  • Definition of linear dependence relation on a set of vectors
  • Definition of trivial linear dependence relation on a set of vectors
  • Determine if a particular set of vectors in R^3 in linearly independent
  • Definition of linearly independent set of vectors: if a linear combination is 0
  • Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors
  • Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors
  • Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0
  • A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.
  • A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.
  • If a set of vectors contains the 0 vector
  • A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.
  • If a set of vectors in R^n (or C^n) contains more than n elements
  • A subset of a linearly independent set is linearly independent.
  • A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
  • Removing a linearly dependent vector from a set does not change the span of the set.
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
  • Any linearly independent set can be expanded to a basis for the (sub)space
KnowenBeta
#
  • Explore
  • Sign in
  1. A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.

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
676–700 of 715 nodes
  • The solutions to a homogeneous linear differential equation is a vector space.#24102
  • The solutions to a homogeneous system of linear equations is a vector space.#24101
  • The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.#23445
  • The span of a set of vectors is a subspace#23977
  • The spectral theorem for symmetric matrices#23575
  • The standard inner product of a vector with itself is 0 only for the 0 vector#23903
  • The standard inner product of a vector with itself is non-negative#23902
  • The standard inner product on C^n can be written as the product of a vector and the adjoint of a vector.#23905
  • The standard inner product on C^n commutes/anticommutes with scalar multiplication.#23900
  • The standard inner product on C^n is anticommutative.#23898
  • The standard inner product on R^n can be written as the product of a vector and the transpose of a vector.#23904
  • The standard inner product on R^n commutes with (real) scalar multiplication.#23899
  • The standard inner product on R^n is commutative.#23897
  • The standard inner product on R^n (or C^n) distributes over addition.#23901
  • The standard/natural basis of R^n (or C^n) is a basis.#23831
  • The sum of linear transformations is a linear transformation#24016
  • The sum of subspaces is a subspace#23969
  • The the image of a spanning set is a spanning set for the range space#24037
  • The transpose of a product of matrices is the product of the transposes in reverse order.#23512
  • The transpose of a sum of matrices is the sum of the transposes.#23510
  • The triangle inequality#23912
  • The union of bases from independent subspaces is a basis for the space.#23974
  • TODO: Linear Algebra Content Progress#29496
  • Transpose and adjoint#23502
  • Transpose commutes with scalar multiplication.#23511
« First ← Previous ... 24 25 26 27 28 29 Next → Last »
1–24 of 24 public trees
  • CTSI Resources#23365
  • 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
  • Misc#929
  • my test article#24378
  • Paid Search#24168
  • Personal Knowledgebases#1202
  • Philosophy#25227
  • Physics#18
  • Products of Knowen#22833
  • Recyling Bin#23013
  • Resources for Mathematics#481
  • Security and Strategic Studies#23169
  • TECHNOLOGY and MARKETS#24392
  • Test#22965
  • Testing#23012
  • test jhjhk#26989
  • User Suggestions and Comments#24157
  • Workshops#1170

Knowen© 2026 KTree Inc.

ExplorePricingIntegrationsContact usLeave feedback