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 homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.

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
476–500 of 715 nodes
  • Linear systems and echelon matrices#23770
  • Linear systems and matrices#23751
  • Linear systems have 0#23439
  • Linear systems of equations#23411
  • Linear transformations#23839
  • LU decomposition#23624
  • Matrices#23447
  • Matrices act as a transformation by multiplying vectors#23855
  • Matrices as linear transformations#23854
  • Matrix addition is commutative and associative.#23463
  • Matrix adjoint is an involution.#23509
  • Matrix conjugation is an involution.#23466
  • Matrix describing a rotation of the plane#23887
  • Matrix diagonalization#23609
  • Matrix equations#23763
  • Matrix equivalence#23603
  • Matrix inverse is an involution.#23522
  • Matrix inverses are unique: if A and B are square matrices#23533
  • Matrix multiplication can be viewed as the dot product of a row vector of column vectors with a column vector of row vectors#23501
  • Matrix multiplication is associative.#23496
  • Matrix multiplication is distributive over matrix addition.#23497
  • Matrix multiplication is not commutative in general.#23499
  • Matrix representation of a composition of linear transformations is given by a matrix product#23862
  • Matrix-scalar multiplication is commutative#23472
  • Matrix-scalar product is commutative#23473
« First ← Previous ... 16 17 18 19 20 21 22 23 24 ... 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