Putting into consideration the polynomials p₁ = 1 + x, p₂ = 1 x² , p₃ = 1 + x + x³ , p₄ = 1 + x³ and p₅= x² + x³

(a) Show, or argue, that the five polynomials {p₁, p₂, p₃, p₄, p₅} are linearly dependent.

(b) Show that the four polynomials {p₁, p₂, p₃, p₄} are linearly independent.

(c) Express p =617 -37x -21x2 - 3x 3 as a linear combination of p₁, p₂, p₃, p₄.

(d) Why can we be certain (without any calculation) that the polynomial f(x) = 1 + x + x2 + x³ can be expressed as a linear combination of the polynomials p₁(x), p₂(x), p₃(x), p₄(x) such that f(x) = ₁p₁(x) + ₂p₂(x) + ₃p₃(x) + ₄p₄(x)? Determine the coefficient ₂ in the expansion above (without solving the linear equation system)?

(e) Assuming the scalar product to be defined as {p,r} = 01p(x) r(x) dx, test if p₁ and p₅ are orthogonal to each other.