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Chemical Engineering Hand Written_Notes_Part_27

# Chemical Engineering Hand Written_Notes_Part_27 - 56 3...

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56 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Figure 1 the following system of equations (1.7) " 2 1 1 1 # x = " 1 5 # There are two ways of interpreting the above matrix vector equation geometri- cally. Row picture : If we consider two equations separately as (1.8) 2 x y = " 2 1 # T " x y # = 1 (1.9) x + y = " 1 1 # T " x y # = 5 then, each one is a line in x-y plane and solving this set of equations simultaneously can be interpreted as fi nding the point of their intersec- tion (see Figure 1 (a)). Column picture : We can interpret the equation as linear combina- tion of column vectors, i.e. as vector addition (1.10) x 1 " 2 1 # + x 2 " 1 1 # = " 1 5 # Thus, the system of simultaneous equations can be looked upon as one vector equation i.e. addition or linear combination of two vectors (see Figure 1 (b)).

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1. SOLUTION OF Ax = b AND FUNDAMENTAL SPACES OF A 57 Now consider the following set of equations (1.11) " 1 2 1 2 # " x 1 x 2 # = " 2 5 # In row picture, this is clearly an inconsistent case( 0 = 1 ) and has no solution as the row vectors are linearly dependent. In column picture, no scalar multiple
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Chemical Engineering Hand Written_Notes_Part_27 - 56 3...

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