theorem 5.43 The four fundamental subspaces

Rests on

• depends_on corollary 5.29 Orthogonal decomposition ¶
  • depends_on definition 5.18 Inner product ¶
    • depends_on definition 4.33 Vector space ¶
      • depends_on definition 4.8 Commutativity; abelian structure ¶
        • depends_on definition 4.4 Internal binary operation; magma ¶
      • depends_on definition 4.32 Field ¶
        • depends_on definition 4.11 Inverse element ¶
        • depends_on definition 4.20 Monoid ¶
        • depends_on definition 4.30 Ring ¶
      • depends_on definition 4.31 Module ¶
        • depends_on definition 4.29 Action ¶
        • depends_on definition 4.28 Distributivity ¶
        • depends_on definition 4.21 Group ¶
        • depends_on definition 4.30 Ring ¶ ↺
  • depends_on definition 5.7 Vector subspace ¶
    • depends_on definition 5.5 Linear combination ¶
      • depends_on definition 4.33 Vector space ¶ ↺
  • depends_on proposition 5.28 Gram–Schmidt ¶
    • depends_on definition 5.14 Linear independence ¶
      • depends_on definition 5.5 Linear combination ¶ ↺
    • depends_on definition 5.18 Inner product ¶ ↺
    • depends_on definition 5.27 Orthonormal basis ¶
      • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.24 Orthogonal vectors ¶
      • depends_on definition 5.25 Unit vector ¶
        • depends_on definition 5.19 Norm ¶
    • depends_on equation 5.44 eq:lin-norm-assoc ¶
    • proves proof ch:03-linear-algebra-representations@proof-5 ¶
  • proves proof ch:03-linear-algebra-representations@proof-6 ¶
• depends_on definition 5.41 Adjoint ¶
  • depends_on definition 5.18 Inner product ¶ ↺
  • depends_on definition 5.37 Linear transformation ¶
    • depends_on definition 4.33 Vector space ¶ ↺
• depends_on theorem 5.40 Rank–nullity ¶
  • depends_on definition 5.15 Basis ¶ ↺
  • depends_on definition 5.39 Kernel, image, nullity, rank ¶
    • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on definition 5.7 Vector subspace ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-12 ¶
  • depends_on lemma 5.38 Exchange and completion ¶
    • depends_on definition 5.15 Basis ¶ ↺
    • depends_on definition 5.12 Subspace generated by a set of vectors ¶
      • depends_on definition 5.5 Linear combination ¶ ↺
      • depends_on definition 5.7 Vector subspace ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-3 ¶
    • depends_on definition 5.14 Linear independence ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-11 ¶
  • proves proof ch:03-linear-algebra-representations@proof-13 ¶
• proves proof ch:03-linear-algebra-representations@proof-15 ¶

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