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I n particular if A and Β are sets t h e n A = Β if and only if every element of t h e set A is also an element of the set Β and conversely, every element of the set Β is an element of t h e set A. 2 SUBSETS If A and Β are sets such that every element of Β is also an element of A then Β is a subset of A or Β is contained in A. T h i s statement is written as Β cz A. Since A is always a subset of itself, if Β a A and A is not equal to Β then Β is a proper subset of A. F o r example consider the following s e t s : A = (a> by c9 dy e)y D= Β = (ay by c)y (aydybyCye)y E= C = (af d, b), (βy f ) here A = D and Β and C are proper subsets of A and D.

F- 12x2 — 4Λ;3 - ( - 4*! + 4Λ2 4*3 + 2x4 16 — 6*4 4 U s i n g C r a m e r ' s rule AI = 4 2 6 4 2 2 12 4 10 - 6 - 4 4 2 - 8 2 - 6 -240 39 9. Linear Equations |Α2| |Α3| AJ 10 - 2 -16 4 2 2 12 4 10 -6 - 4 4 2 -8 2 - 6 -480 = 4 2 6 4 10 - 2 16 4 10 -6 - 4 4 2 -8 2 - 6 -48 = 4 2 6 4 2 2 12 4 10 - 2 16 4 2 - 8 2 - 6 = 4 2 6 4 2 2 12 4 10 -6 - 4 4 10 - 2 16 4 = 0 -192 so that X l = - 4 8 0 / ( - 2 4 0 ) = 2 , x2 = - 4 8 / ( - 2 4 0 ) = 0 . 8. 3 HOMOGENOUS EQUATIONS A linear equation of t h e form (2-59) is called a homogenous equation, A system of homogenous equations can be denoted by Ax = 0 (zero column vector) linear (2-60) Clearly t h e rank of A is always equal to t h e rank of t h e a u g m e n t e d matrix [AO].

T h e statement that a = b means that t h e object a and the object b are identical. I n particular if A and Β are sets t h e n A = Β if and only if every element of t h e set A is also an element of the set Β and conversely, every element of the set Β is an element of t h e set A. 2 SUBSETS If A and Β are sets such that every element of Β is also an element of A then Β is a subset of A or Β is contained in A. T h i s statement is written as Β cz A. Since A is always a subset of itself, if Β a A and A is not equal to Β then Β is a proper subset of A.

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