(b) (c) (d) (e) Let V = C[0, 1] be the space of real-valued continuous functions defined on [0, 1]. Let S be the subset

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B C D E Let V C 0 1 Be The Space Of Real Valued Continuous Functions Defined On 0 1 Let S Be The Subset 1 (85.83 KiB) Viewed 38 times

(b) (c) (d) (e) Let V = C[0, 1] be the space of real-valued continuous functions defined on [0, 1]. Let S be the subset of V comprising all functions f satisfying f(0) = 0 and f'(1) = 0. Determine whether S is a subspace of V. Let Mnn denote the space of all real square matrices of dimension n. Let T denote the transformation T: Mnn → Mnn defined by T(M) = M + B for a fixed matrix B. Determine whether T is a linear transformation. Let T: P₂ → M22 be the linear transformation defined by T (p) [ (i) (ii) p'(0) So p(x) dx Let V be the function space C[-, π]. Consider the set S = {1, cos(x), sin(x), cos(2x), sin(2x)} CV. So p(x) dx p'(1) Find the matrix A for T with respect to the standard bases for both spaces. Show that range(T) = S₂, the set of all 2 × 2 symmetric matrices. Describe a process by which you could prove, using linear algebra techniques, that S is a linearly independent set. You may assume that you can evaluate any of the functions in the set at any arbitrary values of x as part of your solution.