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Chemical Engineering Hand Written_Notes_Part_13

Chemical Engineering Hand Written_Notes_Part_13 - 28 2...

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28 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS Thus, the fundamental property of objects (elements) in a vector space is that they can be constructed by simply adding other elements in the space. This property is formally de f ned as follows. Definition 5 . ( Linear Combination): A linear combination of vectors x (1) , x (2) , , ....... x ( m ) in a vector space is of the form α 1 x (1) + α 2 x (2) + .............. + α m x ( m ) where ( α 1 , ...α m ) are scalars. Note that we are dealing with set of vectors (2.2) © x ( k ) : k =1 , 2 , ...... m. ª .Thu s ,i f X = R n and x ( k ) R n represents k’th vector in the set, then it is a vector with n components i.e. (2.3) x ( k ) = h x ( k ) 1 x ( k ) 2 .... x ( k ) n i T Similarly, if X = l and x ( k ) l represents k’th vector in the set, then x ( k ) represents a sequence of in f nite components (2.4) x ( k ) = h x ( k ) 1 .... x ( k )
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Chemical Engineering Hand Written_Notes_Part_13 - 28 2...

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