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**Unformatted text preview: **C [0 , 1] deﬁne f + g by ( f + g )( x ) = f ( x )+ g ( x ) (we know that then f + g is also continuous) and natural multiplication by scalars: if c ∈ R and f ∈ C [0 , 1] then cf ∈ C [0 , 1] is deﬁned as ( cf )( x ) = cf ( x ). a) Show that C [0 , 1] is a vector space over R . What is its dimension? b) (BONUS extra 10p.) Show that the functions e x , e 2 x , e 3 x are linearly independent in C [0 , 1]. c) Assuming b) is true, let V = Sp ( e x , e 2 x , e 3 x ) subspace of C [0 , 1], and let T : V → V be given by T ( p ) = p ( x ). Show that T is invertible and ﬁnd T-1 . ( Hint: ﬁnd T-1 on a basis, then extend by linearity to its domain V .) 1...

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