Please answer all questions inorder to gett full point
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 R3 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 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.