Question

Encuentre las coordenadas del vector x relativas a la base ortonormal B y realice una grafica mostrando los resultados

{(√5/5,(2√5)/5),(-(2√5)/5,√5/5)},X=(-3,4)

Answer 1

Sea (a,b) las coordenadas de X = (-3,4) en la base $\left\{\left(\dfrac{\sqrt5}{5},\dfrac{2\sqrt5}{5}\right),\left(-\dfrac{2\sqrt5}{5},\dfrac{\sqrt5}{5}\right)\right\}$ entonces tenemos

$a\left(\dfrac{\sqrt5}{5},\dfrac{2\sqrt5}{5}\right)+b\left(-\dfrac{2\sqrt5}{5},\dfrac{\sqrt5}{5}\right)=(-3,4)\\ \\ \\\begin{cases}\dfrac{\sqrt5}{5}a-\dfrac{2\sqrt5}{5}b=-3\\ \\\dfrac{2\sqrt5}{5}a+\dfrac{\sqrt5}{5}b=4\end{cases}\\ \\ \\a=\sqrt{5}~~ \wedge ~~b=2\sqrt{5}\\ \\ \\\text{Entonces las coordenadas son }\left(\sqrt{5},2\sqrt{5}\right)$