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Let r R and For which values r R is the set {u. v. w} linearly independent? For which values r R is the vector b a linear combination of u, v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set R of all column vectors of length three, with real entries, is a vector space. Is the subset a subspace of R3? Justify your answer. Show that the set of all twice differentiable functions f : R rightarrow R satisfying the differential equation sin (x)f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space P3 of all real polynomials p of degree at most 3: S = {p P3 | p(1) = 0, p'(1) = 1}, where p' is the derivative of p. Determine whether S is a subspace of P3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S.
