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Solution of Linear Equations

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Linear Equations

Let _ A#{~x} = #{~b} _ represent a system of ~m equations in ~n unknowns. _ [i.e. A is a ~m # ~n matrix, #{~x} is a ~1 # ~n and #{~b} a ~1 # ~m column vector.]

matrix{~a_{1 1}, ... ,~a_{1 ~n} /., ... ,./~a_{~m 1}, ... ,~a_{~m ~n}} matrix{~x_{1}/ ... /~x_{~n}} _ = _ matrix{~b_{1}/ ... /~b_{~m}}

So

array{~a_{1 1} ~x_{1} _ + _ ... _ + _ ~a_{1 ~n} ~x_{~n}, _ = _ ,~b_{1} / . _ . _ . _ . _ . _ . _ ,,. / . _ . _ . _ . _ . _ . _ ,,. /~a_{~m 1} ~x_{1} _ + _ ... _ + _ ~a_{~m ~n} ~x_{~n} , _ = _ , ~b_{~m} }

Solution Conditions

If _ A#{~x} = #{~b} _ is a system of ~m equations in ~n unknowns

  1. A#{~x} = #{~b} _ has a solution (at least one) _ <=> _ rank(A) = rank(A|#{~b})
  2. A#{~x} = #{~b} _ has a unique solution _ <=> _ rank(A) = ~n
  3. If A is ~n # ~n _ A#{~x} = #{~b} _ has a unique solution _ <=> _ |A| ≠ 0
  4. A#{~x} = #{~b} _ and _ A#{~z} = #0 _ &imply. _ A(#{~x + λ~z} ) = #{~b} _ &forall. λ
  5. A#{~z} = #0 _ has a non-trivial solution _ <=> _ rank(A) < ~n
  6. If A is ~n # ~n _ A#{~z} = #0 _ has a non-trivial solution _ <=> _ |A| = 0

Cramer's Rule

If _ A#{~x} = #{~b} _ is a system of ~n equations in ~n unknowns and |A| ≠ 0 _ then _ the solution of the equation is given by

~x_{~i} _ = _ fract{&Delta._{~i},|A|} _ _ _ _ _ _ 1 &le. ~i &le. ~n

where &Delta._{~i} is the determinant of the matrix obtained by replacing the ~i^{th} column of A by #{~b}.