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   DIR Return to: Algebra
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       #Post#: 71--------------------------------------------------
       Linear Algebra
   DIR By: Dragon Hellfire
       Date: March 22, 2019, 6:12 am
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       Operators
       (1) + : Matrix addition
       (2) * : Matrix multiplication
       (3) ' : Transpose
       (4) 0 : Additive identity
       (5) 1 : Multiplicative identity
       Special Matrices
       I := Identity matrix
       P := Pemutation matrix
       E := Elimination matrix
       L := Lower triangular matrix
       U := Upper triangular matrix
       Axioms
       Additive Commutativity:
       A + B = B + A
       Additive Associativity:
       (A + B) + C = A + (B + C)
       Multiplicative Associativity:
       (A*B)*C = A*(B*C)
       Distributivity:
       A*(B+C) = AB + AC
       Transpose Rules
       (A')' = A
       (A+B)' = A' + B'
       (AB)' = B'A'
       Inverse Rules
       A * A^(-1) = I
       A * B * B^(-1) * A^(-1) = I
       (A')^(-1)=(A^(-1))'
       Requirements of Vector Spaces
       (1) Additive commutativity
       (2) Additive associativity
       (3) Additive identity
       (4) ∃ Additive inverse ∀ X
       (5) Scalar associativity
       (6) Distributivity
       (7) Multiplicative identity
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