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Task number: 4374

Determine parameters \( a, b \in \mathbb K \) so that for each quadratic form \( g \) at \( V \) above \( \mathbb K \) and any three vectors \( \mathbf u, \mathbf v, \mathbf w \in V \) holds: \(g(\mathbf u+\mathbf v+\mathbf w)=ag(\mathbf u+\mathbf v)+ag(\mathbf u+\mathbf w)+ag(\mathbf v+\mathbf w)+bg(\mathbf u)+bg(\mathbf v)+bg(\mathbf w)\)

  • Řešení

    We expend both sides using a bilinear form \(f\) and compare the coefficients. \[ \begin{eqnarray*} && g(\mathbf u+\mathbf v+\mathbf w) \\ &=& f(\mathbf u+\mathbf v+\mathbf w,\mathbf u+\mathbf v+\mathbf w)\\ &=& f(\mathbf u,\mathbf u)+ f(\mathbf u,\mathbf v)+ f(\mathbf u,\mathbf w)+ f(\mathbf v,\mathbf u)+ f(\mathbf v,\mathbf v)+ f(\mathbf v,\mathbf v)+ f(\mathbf w,\mathbf u)+ f(\mathbf w,\mathbf v)+ f(\mathbf w,\mathbf w)\\ \end{eqnarray*} \] \[ \begin{eqnarray*} && a(g(\mathbf u+\mathbf v)+g(\mathbf u+\mathbf w)+g(\mathbf v+\mathbf w))+b(g(\mathbf u)+g(\mathbf v)+g(\mathbf w))\\ &=& a(f(\mathbf u+\mathbf v,\mathbf u+\mathbf v)+f(\mathbf u+\mathbf w,\mathbf u+\mathbf w)+f(\mathbf v+\mathbf w,\mathbf v+\mathbf w))+b(f(\mathbf u,\mathbf u)+f(\mathbf v,\mathbf v)+f(\mathbf w,\mathbf w)) \\ &=& (2a+b)(f(\mathbf u,\mathbf u)+ f(\mathbf v,\mathbf v)+ f(\mathbf w,\mathbf w))\\ &&+a( f(\mathbf u,\mathbf v)+ f(\mathbf u,\mathbf w)+ f(\mathbf v,\mathbf u)+ f(\mathbf v,\mathbf v)+ f(\mathbf v,\mathbf v)+ f(\mathbf w,\mathbf u)+ f(\mathbf w,\mathbf v)) \\ \end{eqnarray*} \] Therefore \(2a+b=1\) and \(a=1\).
  • Výsledek

    The coefficients are \(a=1\) and \(b=-1\).
Difficulty level: Easy task (using definitions and simple reasoning)
Reasoning task
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