Find the representation of a vector (either real-space or from a function space) with respect to a set of given orthogonal basis vectors.

(a) Consider the basis for R³ consisting of the vectors r₁ = (1, ?1, 0), r₂ = (?3, ?3, 0) and r₃ = (0, 0, 2). Convince yourself that these vectors are pairwise orthogonal and linearly independent.

(b) Express the vector f = (?3.25, ?4.25, 2) as the linear combination f =?13? c_kr_k . Determine the coefficients c_k by solving the relevant system of linear equations.

(c) Now perform the same task, but by using orthogonality of the basis vectors instead of solving the linear system of equations. Convince yourself that you obtain the result.

(d) Consider the first three elements of the so-called Laguerre polynomials. These are the functions L₁(x) = 1, L₂(x) = ?x + 1 and L₃ = 0.5(x² ? 4x + 2). Assume further that the function s(x) = ?x ? 3x2 + 1 is in the span of these functions, s ? span(L₁, L₂, L₃). Determine the coefficients ?i such that s(x) = ?₁L₁(x) + ?₂L₂(x) + ?₃L₃(x) by solving the corresponding system of equations.

(e) The Laguerre polynomials above are an orthogonal set of functions with respect to the following inner product defined for two functions g(x) and f(x)