Determine a matriz associada a transformação T_{i} com relação as bases \alpha_{i} e \beta_{i}.
T_{2}: M_{2}(R) \rightarrow R^{3}, \quad T_{2}\binom{x y}{z w}=(x, x-3 y+w, z-w) \quad, \quad \beta_{2}=\{(1,1,0),(0,1,0),(0,0,-2)\} \quad e
\alpha_{2}=\left\{\left(\begin{array}{cc} 1 & 0 \\\\ -1 & 0 \end{array}\right),\left(\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\\\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\\\ 0 & 1 \end{array}\right)\right\} .

Passo 1

O enunciado define a transformação linear T_2\colon M_2(\mathbb{R})\to\mathbb{R}^3 dada por
T_2\begin{pmatrix}x&y\z&w\end{pmatrix}=(x,\;x-3y+w,\;z-w).
As bases são
\alpha_2=\left\{\begin{pmatrix}1&0\\\ -1&0\end{pmatrix},\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}\right\},
\beta_2=\{(1,1,0),\,(0,1,0),\,(0,0,-2)\}.
Vamos determinar a matriz de T_2 relativa a \alpha_2 e \beta_2.

Passo 2

Para cada vetor da base \alpha_2 calculamos sua imagem por T_2:
A_1=\begin{pmatrix}1&0\\\ -1&0\end{pmatrix}\implies T_2(A_1)=(1,\;1,\;-1),
A_2=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\implies T_2(A_2)=(0,\;-3,\;-1),
A_3=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\implies T_2(A_3)=(0,\;0,\;1),
A_4=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}\implies T_2(A_4)=(0,\;1,\;-1).

Passo 3

Para escrever cada T_2(A_j) na base \beta_2 resolvemos
(a,b,c)_\beta\text{ tal que }a\,(1,1,0)+b\,(0,1,0)+c\,(0,0,-2)=(v_1,v_2,v_3).
Isso dá
a=v_1,\quad b=v_2-v_1,\quad c=-\frac{v_3}{2}.

Passo 4

Aplicando a fórmula acima:
  • para T_2(A_1)=(1,1,-1) obtemos (a,b,c)=(1,0,\tfrac12),
  • para T_2(A_2)=(0,-3,-1) obtemos (0,-3,\tfrac12),
  • para T_2(A_3)=(0,0,1) obtemos (0,0,-\tfrac12),
  • para T_2(A_4)=(0,1,-1) obtemos (0,1,\tfrac12).

Passo 5

A matriz de T_2 relativa a \alpha_2,\beta_2 tem colunas iguais a esses vetores coordenados, isto é
[T_2]_{\beta_2,\alpha_2}= \begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & -3 & 0 & 1 \\\ \tfrac12 & \tfrac12 & -\tfrac12 & \tfrac12 \end{pmatrix}.

Resposta

A matriz associada é
[T_2]_{\beta_2,\alpha_2}=\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & -3 & 0 & 1 \\\ \tfrac12 & \tfrac12 & -\tfrac12 & \tfrac12 \end{pmatrix}.

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