Chemical Engineering Hand Written_Notes_Part_25

Chemical Engineering Hand Written_Notes_Part_25 - 52 2...

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52 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS (3) Show that functions 1, exp(t), exp(2t), exp(3t) are linearly independent over any interval [a,b]. (4) Does the set of functions of the form f ( t )=1 / ( a + bt ) constitute a linear vector space? (5) Give an example of a function which is in L 1 [0 , 1] but not in L 2 [0 , 1] . (6) Decide linear dependence or independence of (a) (1,1,2), (1,2,1), (3,1,1) (b) ¡ x (1) x (2) ¢ , ¡ x (2) x (3) ¢ , ¡ x (3) x (4) ¢ , ¡ x (4) x (1) ¢ for any x (1) , x (2) , x (3) , x (4) (c) (1,1,0), (1,0,0), (0,1,1), (x,y,z) for any scalars x,y,z (7) Describe geometrically the subspaces of R 3 spanned by following sets (a) (0,0,0), (0,1,0), (0,2,0) (b) (0,0,1), (0,1,1), (0,2,0) (c) all six of these vectors (d) set of all vectors with positive components (8) Consider the space X of all n × n matrices. Find a basis for this vector space and show that set of all lower triangular n × n matrices forms a subspace of X. (9) Determine which of the following de
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Chemical Engineering Hand Written_Notes_Part_25 - 52 2...

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