Seja \varphi: \mathbb{R}^{3} \rightarrow \mathbb{R} a forma quadratica dada por \varphi(x, y, z)=x^{2}+y^{2}-z^{2}+2 x y-3 x z+y z. Qual é matriz de \varphi na base \{u, v, w\} \subset \mathbb{R}^{3}, onde u=(3,0,1), v=(1,-1,2) e w=(2,1,2) ?

Passo 1

O enunciado dá a forma quadrática \varphi:\mathbb{R}^3\to\mathbb{R} definida por
\varphi(x,y,z)=x^{2}+y^{2}-z^{2}+2xy-3xz+yz
e nos pede a matriz de \varphi na base B=\{u,v,w\}, onde
u=(3,0,1),\quad v=(1,-1,2),\quad w=(2,1,2).
Para resolver, vamos (1) achar a matriz simétrica A de \varphi na base canônica e (2) calcular a matriz na base B via
M_B=\bigl[\varphi(u_i,u_j)\bigr]_{i,j} \;=\;\bigl[u_i^{T}A\,u_j\bigr]_{i,j}.

Passo 2

Determinar a matriz A em coordenadas canônicas. Como \varphi(x,y,z)=[x\;y\;z]\,A\,[x\;y\;z]^{T} e os coeficientes são
  • x^{2} com coeficiente 1 A_{11}=1,
  • y^{2} com coeficiente 1 A_{22}=1,
  • -z^{2} A_{33}=-1,
  • 2xy A_{12}=A_{21}=1,
  • -3xz A_{13}=A_{31}=-\tfrac32,
  • yz A_{23}=A_{32}=\tfrac12.
    Logo,
A=\begin{pmatrix} 1 & 1 & -\tfrac32\\\\ 1 & 1 & \tfrac12\\\\ -\tfrac32 & \tfrac12 & -1 \end{pmatrix}.

Passo 3

Calcular A\,u, A\,v e A\,w.
Para u=(3,0,1):
A\,u =\begin{pmatrix}1&1&-\tfrac32\\\ 1&1&\tfrac12\\\ -\tfrac32&\tfrac12&-1\end{pmatrix} \begin{pmatrix}3\\\ 0\\\ 1\end{pmatrix} =\begin{pmatrix}\tfrac32\\[6pt]\tfrac72\\[4pt]-\tfrac{11}2\end{pmatrix}.
Para v=(1,-1,2):
A\,v =\begin{pmatrix}-3\\\ 1\\\ -4\end{pmatrix}.
Para w=(2,1,2):
A\,w =\begin{pmatrix}0\\\ 4\\\ -\tfrac92\end{pmatrix}.]

Passo 4

Construir a matriz em B cujos elementos são
m_{ij}=u_i^{T}A\,u_j.
Assim,
m_{11}=u^{T}(A\,u) =(3,0,1)\cdot\bigl(\tfrac32,\tfrac72,-\tfrac{11}2\bigr)=-1,
m_{12}=u^{T}(A\,v) =(3,0,1)\cdot(-3,1,-4)=-13,
m_{13}=u^{T}(A\,w) =(3,0,1)\cdot(0,4,-\tfrac92)=-\tfrac92,
m_{22}=v^{T}(A\,v) =(1,-1,2)\cdot(-3,1,-4)=-12,
m_{23}=v^{T}(A\,w) =(1,-1,2)\cdot(0,4,-\tfrac92)=-13,
m_{33}=w^{T}(A\,w) =(2,1,2)\cdot(0,4,-\tfrac92)=-5.

Resposta

A matriz de \varphi na base \{u,v,w\} é
\begin{pmatrix} -1 & -13 & -\tfrac92\\\\ -13 & -12 & -13\\\\ -\tfrac92 & -13 & -5 \end{pmatrix}.

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