Revision #906 → #1526 · back to history
addedRank–nullity theorem (statement)e39a3f5dd2b9
addedInjectivity/surjectivity implies bijectivity76bac8c3c27c
addedRank–nullity for linear transformations71483c5efc72
addedRefinement via splitting lemma (isomorphism form)855a781f364c
addedRank–nullity for matricesc7ecb13a47ca
addedFirst proof (general case via linear maps)13fc1852cd9a
addedSecond proof (homogeneous system)cda54e4ac82e
addedCodomain need not be finite-dimensional5f3ada52c176
addedExtension of kernel basis to full basisef73fdd988ee
addedImages of extension vectors form a basis of image0a0b8c787428
addedLinear independence of image vectorsa486130f91c8
addedExistence of n−r linearly independent null-space solutionse550fe11a84f
addedEvery solution is a linear combination of these solutions4432b70824d8
addedConstruction of null-space basis matrix N3f2608d54c68
addedRank factorization of dependent columns1bd193b3e407
addedColumns of N are particular solutions735e16ed767f
addedColumns of N are linearly independentf8fae1bce924
addedCokernel of a linear transformationad018f91775f
addedFundamental theorem of linear algebra7aa28069ddd3
addedEquivalence with first isomorphism theoremc5d4817cc01a
addedShort exact sequences of vector spaces split98e95d96bdd5
addedAlternating sum of dimensions in an exact sequencee7ba1ded2ad4
addedIndex of a linear mapcf7a045faadc
addedRank–nullity as index identityab5e81018602
addedAtiyah–Singer index theorem6b8432b6d031