Let V=P3(x)V = P_3(x)V=P3(x) be a subspace of P(x)P(x)P(x) with inner product ⟨f,g⟩=∫−11f(x)g(x) dx\langle f,g \rangle = \int_{-1}^1 f(x)g(x)\,dx⟨f,g⟩=∫−11f(x)g(x)dx.
(A) Find the matrix AAA with respect to the basis {1,x,x2,x3}\{1,x,x^2,x^3\}{1,x,x2,x3} in VVV such that ⟨f,g⟩=[f]TA[g]\langle f,g \rangle = [f]^T A [g]⟨f,g⟩=[f]TA[g].
(B) Find the Fourier coefficient of x5x^5x5 along x3−35xx^3 - \frac{3}{5}xx3−53x.
參考答案與解析
(A) 內積矩陣 A=[202/3002/302/52/302/5002/502/7]A=\begin{bmatrix}2&0&2/3&0\\0&2/3&0&2/5\\2/3&0&2/5&0\\0&2/5&0&2/7\end{bmatrix}A=202/3002/302/52/302/5002/502/7，其中Aij=⟨xi,xj⟩=∫−11xi+jdxA_{ij}=\langle x^i,x^j\rangle=\int_{-1}^1x^{i+j}dxAij=⟨xi,xj⟩=∫−11xi+jdx，當i+ji+ji+j為奇數時積分為0，偶數時為2i+j+1\dfrac{2}{i+j+1}i+j+12（i,j=0,1,2,3i,j=0,1,2,3i,j=0,1,2,3對應基底1,x,x2,x31,x,x^2,x^31,x,x2,x3）。
(B) 傅立葉係數為 109\dfrac{10}{9}910。計算⟨x5,x3−35x⟩=29−35⋅27=16315\langle x^5,x^3-\frac35x\rangle=\frac{2}{9}-\frac35\cdot\frac27=\frac{16}{315}⟨x5,x3−53x⟩=92−53⋅72=31516，⟨x3−35x,x3−35x⟩=27−65⋅25+925⋅23=8175\langle x^3-\frac35x,x^3-\frac35x\rangle=\frac27-\frac65\cdot\frac25+\frac9{25}\cdot\frac23=\frac{8}{175}⟨x3−53x,x3−53x⟩=72−56⋅52+259⋅32=1758，係數=16/3158/175=109=\frac{16/315}{8/175}=\frac{10}{9}=8/17516/315=910。
