Question

quais são as coordenadas do vetor v =( -3,2 ) em relação a base b = {( raiz de 3 ,1) , ( raiz de 3 , -1 )}

Answer 1

$B=\{(\sqrt{3},1);(\sqrt{3},-1)\}$

$\lambda_1(\sqrt{3},1)+\lambda_2(\sqrt{3},-1)=(-3,2)$

en forma matricial

$\left[\begin{matrix} \sqrt{3}&1\\ \sqrt{3}&-1 \end{matrix}\right] \left[\begin{matrix} \lambda_1\\ \lambda_2 \end{matrix}\right]= \left[\begin{matrix} -3\\ 2 \end{matrix}\right]$

Resolviendo...

$\lambda_1=\frac{\left|\begin{matrix} -3&1\\ 2&-1 \end{matrix}\right|}{\left|\begin{matrix} \sqrt{3}&1\\ \sqrt{3}&-1 \end{matrix}\right|}=-\frac{1}{2\sqrt{3}}$


$\lambda_2=\frac{\left|\begin{matrix} \sqrt{3}&-3\\ \sqrt{3}&2 \end{matrix}\right|}{\left|\begin{matrix} \sqrt{3}&1\\ \sqrt{3}&-1 \end{matrix}\right|}=-\frac{5}{2}$

Entonces el vector en la base B es

                $\vec v_B=\left(-\frac{1}{2\sqrt{3}},-\frac{5}{2}\right)$