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2.1. Show that all solutions of a linear, homogeneous and nth order ordina differential equation constitute an n-dimensional linear vector space.
2.5. The linear vector space formed by all real-valued functions which are contin- uous on the interval -1<x< 1 is denoted by C[ - 1,1). Which of the following subsets of C[ - 1,1) are subspaces: (i) The set of all differentiable functions; (ii) the set of all polynomials of degree n; (iii) the set of all even functions [ f(x) = f(-x) for all x); (iv) the set of all odd functions If(x) = -f(-x) for all x]; (v) the set of all functions f with f(0) = 0; (vi) the set of all functions f with f(0) = 1; (vii) the set of all non-negative functions (f (3) 0 for all x). 2.6. Show that the space C[-1, 1) defined in the preceding exercise is the direct sum of the subspaces determined by the descriptions (iii) and (iv).