Question

Sean B = {(2,−1),(1,1)} y D = {x^2,x + 1,x−1} bases de R2 y R2[x] respectivamente. Determinar [F]D,B, si F : R2 →R2[x] es una T.L. tal que F (2,−1) = x2 −1 yF (3,0) = x^2 + x + 2.

Answer 1

                $F(2,-1)=x^2-1\\ F(3,0)=x^2+x+2\\ \\ \texttt{Faltar\'ia encontrar }F(1,1)\texttt{, pongamos que sea:}\\ \\ F(1,1)=aF(2,-1)+bF(3,0)\\ \\ F(1,1)=a(x^2-1)+b(x^2+x+2)\\ \\ F(1,1)=(a+b)x^2+bx-a+2b\\ \\ \texttt{Por otro lado: }\\ \\ F(1,1)=aF(2,-1)+bF(3,0)\\ \\ F(1,1)=F(2a,-a)+F(3b,0)\\ \\ F(1,1)=F(2a+3b,-a)\\ \\ 2a+3b=1\wedge a=-1 \to a=-1\wedge b=1\\ \\ F(1,1)=x+3\\ \\ F(p(2,-1)+q(1,1))=p(x^2-1)+q(x+3)\\ \\ F(p(2,-1)+q(1,1))=px^2+qx+(3q-p)$


                 $F(p(2,-1)+q(1,1))=px^2+\left(\dfrac{4q-p}{2}\right)(x+1)+\left(\dfrac{p-2q}{2}\right)(x-1)$