91影视

An invertible linear transformation T is called an isomorphism.

Also says that the linear space V is isomorphic to the linear space W if there exists an isomorphism T from V to W.

Consider a basis $f_1,\mathrm{...},f_n$of $P_{n-1}$.

Let $a_1,\mathrm{...},a_n$be distinct real numbers.

Consider the n 脳 n matrix M whose ijth entry is $f_j\left(a_i\right)$.

.Here also given that the value of the weights are a₁=-1,a₂=0,a₃=1

Assume that the basis $B=\left(f_1,\mathrm{...},f_n\right)$is the basis of a linear space V then the coordinate transformation $L_B\left(f\right)={\left[f\right]}_B$ from V to Rⁿ is an isomorphism

Now when applying simpson鈥檚 rule in calculus that there exist 鈥渨eights鈥� $w_1,\mathrm{...},w_n$such that $\underset{-1}{\overset{1}{鈭�}}f\left(t\right)dt=\underset{i=1}{\overset{n}{鈭�}}{w}_{i}f\left(a_i\right)$.

Here by the properties of the isomorphisms we have that the weights $w_1,\mathrm{...},w_n$obtains the function $\underset{-1}{\overset{1}{鈭�}}f\left(t\right)dt=\underset{i=1}{\overset{n}{鈭�}}{w}_{i}f\left(a_i\right)$for all the polynomials in $P_{n-1}$.

Hence the proof.