linear independence
No redundant vectors
Linear independence is a mathematical property of a set of vectors: no vector in the set can be expressed as a combination of the others.
If you can write one vector as a sum of multiples of the rest, the set is linearly dependent.
The concept is foundational in linear algebra, determining whether vectors span a space efficiently or contain redundancy.
The three vectors are linearly independent, so they form a basis for three-dimensional space.
Subset forming smaller space
- dimensionMeasurable extent or scope
- transformationThorough change in form
- isomorphismStructure-preserving bijective map
- kernelSeed or essential core
Alike in composition throughout
- system of equationsMultiple equations sharing variables
- consistentNot varying or contradicting
- overdeterminedHaving excess determining factors
- underdeterminedHaving insufficient constraints
solvability · parametric · simultaneous equations · constraint
Position in hierarchy
- pivotPoint of rotation
- kernelSeed or essential core
- determinantMatrix-derived scalar quantity
- invertibleAble to be undone
Distance between endpoints
- orthogonalityState of being independent
- Gram-SchmidtVector orthogonalization algorithm
- coordinate systemMethod for locating points
- dimensionMeasurable extent or scope
State of being independent
- dot productSum of component products
- eigenvectorDirection preserved under transformation
- coplanarExisting in one plane
- parallelEquidistant at all points
perpendicular · component · magnitude · direction · cross product · projection
subspace
Subset forming smaller space
- dimensionMeasurable extent or scope
- transformationThorough change in form
- isomorphismStructure-preserving bijective map
homogeneous
Alike in composition throughout
- system of equationsMultiple equations sharing variables
- consistentNot varying or contradicting
- overdeterminedHaving excess determining factors
underdetermined · solvability · parametric · simultaneous equations · constraint
rank
Position in hierarchy
- pivotPoint of rotation
- kernelSeed or essential core
- determinantMatrix-derived scalar quantity
span
Distance between endpoints
- orthogonalityState of being independent
- Gram-SchmidtVector orthogonalization algorithm
- coordinate systemMethod for locating points
orthogonality
State of being independent
- dot productSum of component products
- eigenvectorDirection preserved under transformation
- coplanarExisting in one plane
parallel · perpendicular · component · magnitude · direction · cross product · projection
- linear algebramatrix, vectors, matrices, determinant, variables
- vector spacesvector, basis, dimension, subspace, eigenvector
- matricestranspose, determinants, rows, cells, Hermitian matrices
- ring theoryrings, Noetherian rings, ideal, ideals, quotient ring
- tensor algebratensors, vectors, scalars, components, dyads
linear independencedependence
State of needing another
collinearity · redundancy · dependency · derivability · span
linear independencedegeneracy
Moral or physical decline
collinear · singularity · coplanar · redundant · derivable
independence, linear, nonlinear, linearity, collinear, linearly, linearize, nonlinearity, interlinear