申論題

四、 (15%,計算題) Let P₃ = {c₀ + c₁x+ c₂x + c₃x³}, be the set of all n-th order polynomials with real-valued ci. Define the inner product of two vectors in P₃, say u = c0 +... + c₃x³ and v= d₀+... + d₃x³, as ＜u, v＞ c₀d₀ + c₁d₁ + c₂d₂ + c₃ d₃. Note that P₃ can be regarded as a normed vector space.

(D).(3%) Use the set {1, x, x², x³} as the basis for P₃. Find the matrix representation of the linear transformation that does differentiation on polynomials in P₃.