Solve the problem.
-The columns of $\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$ are $\mathbf { e } _ { 1 } = \left[ \begin{array} { l } 1 \ 0 \ 0 \end{array} \right] , \mathbf { e } _ { 2 } = \left[ \begin{array} { l } 0 \ 1 \ 0 \end{array} \right] , \mathbf { e } _ { 3 } = \left[ \begin{array} { l } 0 \ 0 \ 1 \end{array} \right]$.
Suppose that $\mathrm { T }$ is a linear transformation from $R ^ { 3 }$ into $\mathcal { R } ^ { 2 }$ such that
$\mathrm { T } \left( \mathbf { e } _ { 1 } \right) = \left[ \begin{array} { r } 3 \ - 2 \end{array} \right] , \mathrm { T } \left( \mathbf { e } _ { 2 } \right) = \left[ \begin{array} { l } 5 \ 0 \end{array} \right]$, and $\mathrm { T } \left( \mathbf { e } _ { 3 } \right) = \left[ \begin{array} { r } - 5 \ 1 \end{array} \right]$
Find a formula for the image of an arbitrary $x = \left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \end{array} \right]$ in $R ^ { 3 }$.
A)
$T \left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } - 2 x _ { 2 } \ 5 x _ { 1 } \end{array} \right]$
B)
$T \left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } - 2 x _ { 2 } \ 5 x _ { 1 } \ 5 x _ { 2 } + x _ { 3 } \end{array} \right]$
C)
$$T \left[ \begin{array} { l }
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right] = \left[ \begin{array} { l }
3 x _ { 1 } + 5 x _ { 2 } - 5 x _ { 3 } \
5 x _ { 1 } \
- 2 x _ { 1 } + x _ { 3 }
\end{array} \right]$$
D)
$$T \left[ \begin{array} { l }
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right] = \left[ \begin{array} { l l }
3 x _ { 1 } + 5 x _ { 2 } - 5 x _ { 3 } \
- 2 x _ { 1 } + x _ { 3 }
\end{array} \right]$$

Question

Solve the problem.
-The columns of $\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]$ are $\mathbf { e } _ { 1 } = \left[ \begin{array} { l } 1 \ 0 \ 0 \end{array} \right] , \mathbf { e } _ { 2 } = \left[ \begin{array} { l } 0 \ 1 \ 0 \end{array} \right] , \mathbf { e } _ { 3 } = \left[ \begin{array} { l } 0 \ 0 \ 1 \end{array} \right]$.
Suppose that $\mathrm { T }$ is a linear transformation from $R ^ { 3 }$ into $\mathcal { R } ^ { 2 }$ such that
$\mathrm { T } \left( \mathbf { e } _ { 1 } \right) = \left[ \begin{array} { r } 3 \ - 2 \end{array} \right] , \mathrm { T } \left( \mathbf { e } _ { 2 } \right) = \left[ \begin{array} { l } 5 \ 0 \end{array} \right]$, and $\mathrm { T } \left( \mathbf { e } _ { 3 } \right) = \left[ \begin{array} { r } - 5 \ 1 \end{array} \right]$
Find a formula for the image of an arbitrary $x = \left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \end{array} \right]$ in $R ^ { 3 }$.
A)
$T \left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } - 2 x _ { 2 } \ 5 x _ { 1 } \end{array} \right]$
B)
$T \left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } - 2 x _ { 2 } \ 5 x _ { 1 } \ 5 x _ { 2 } + x _ { 3 } \end{array} \right]$
C)
$$T \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right] = \left[ \begin{array} { l } 
3 x _ { 1 } + 5 x _ { 2 } - 5 x _ { 3 } \
5 x _ { 1 } \
- 2 x _ { 1 } + x _ { 3 }
\end{array} \right]$$
D)
$$T \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right] = \left[ \begin{array} { l l } 
3 x _ { 1 } + 5 x _ { 2 } - 5 x _ { 3 } \
- 2 x _ { 1 } + x _ { 3 }
\end{array} \right]$$

Quizplus · Accepted Answer

The correct formula for the image of an arbitrary vector $x$ under the transformation $T$ is obtained by applying $T$ to each basis vector (the columns of the identity matrix) and then combining these images according to the coefficients $x_1, x_2, x_3$ in the vector $x$. This results in a linear combination of the images of the basis vectors, which is exactly what option D describes.

Solve the Problem I3=[100010001]\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]I3​=​100​010​001​​

Solve the Problem $\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$