$$\text { Find the matrix of the linear transformation } \mathrm { T } : \mathrm { V } \rightarrow \mathrm { W } \text { relative to } \mathrm { B } \text { and } \mathrm { C } \text {. }$$
-Suppose $B = \left\{ b _ { 1 } , b _ { 2 } \right\}$ is a basis for $V$ and $C = \left\{ c _ { 1 } , c _ { 2 } , c _ { 3 } \right\}$ is a basis for $W$. Let $T$ be defined by
$$\begin{array} { l }
\mathrm { T } \left( \mathbf { b } _ { 1 } \right) = 5 c _ { 1 } + 6 c _ { 2 } - 5 c _ { 3 } \
T \left( b _ { 2 } \right) = 5 c _ { 1 } + 12 c _ { 2 } + 7 c _ { 3 }
\end{array}$$
A)
$\left[\begin{array}{rr}
5 & 5 \
6 & 12 \
-5 & 7
\end{array}\right]$

B)
$$\left[ \begin{array} { r r }
5 & 5 \
6 & 12 \
- 5 & 7
\end{array} \right]$$

C)
$\left[\begin{array}{rr}
5 & 0 \
6 & 6 \
-5 & 12
\end{array}\right]$

D)
$\left[\begin{array}{rrr}
5 & 6 & -5 \
5 & 12 & 7
\end{array}\right]$

Question

$$\text { Find the matrix of the linear transformation } \mathrm { T } : \mathrm { V } \rightarrow \mathrm { W } \text { relative to } \mathrm { B } \text { and } \mathrm { C } \text {. }$$
-Suppose $B = \left\{ b _ { 1 } , b _ { 2 } \right\}$ is a basis for $V$ and $C = \left\{ c _ { 1 } , c _ { 2 } , c _ { 3 } \right\}$ is a basis for $W$. Let $T$ be defined by
$$\begin{array} { l } 
\mathrm { T } \left( \mathbf { b } _ { 1 } \right) = 5 c _ { 1 } + 6 c _ { 2 } - 5 c _ { 3 } \
T \left( b _ { 2 } \right) = 5 c _ { 1 } + 12 c _ { 2 } + 7 c _ { 3 }
\end{array}$$
A)
$\left[\begin{array}{rr}
5 & 5 \
6 & 12 \
-5 & 7
\end{array}\right]$

B)
$$\left[ \begin{array} { r r } 
5 & 5 \
6 & 12 \
- 5 & 7
\end{array} \right]$$

C)
$\left[\begin{array}{rr}
5 & 0 \
6 & 6 \
-5 & 12
\end{array}\right]$

D)
$\left[\begin{array}{rrr}
5 & 6 & -5 \
5 & 12 & 7
\end{array}\right]$

Quizplus · Accepted Answer

The matrix of the linear transformation $T$ relative to $B$ and $C$ is$$\left[\begin{array}{rr}5 & 5 \6 & 12 \-5 & 7\end{array}\right]$$because$$\begin{aligned}\mathrm { T } \left( \mathbf { b } _ { 1 } \right) &=5 c _ { 1 } + 6 c _ { 2 } - 5 c _ { 3 } \&=5\left[ \begin{array} { l } 1 \0 \0\end{array} \right] + 6\left[ \begin{array} { l } 0 \1 \0\end{array} \right] - 5\left[ \begin{array} { l } 0 \0 \1\end{array} \right] \&=\left[ \begin{array} { r } 5 \6 \-5\end{array} \right] \\mathrm { T } \left( \mathbf { b } _ { 2 } \right) &=5 c _ { 1 } + 12 c _ { 2 } + 7 c _ { 3 } \&=5\left[ \begin{array} { l } 1 \0 \0\end{array} \right] + 12\left[ \begin{array} { l } 0 \1 \0\end{array} \right] + 7\left[ \begin{array} { l } 0 \0 \1\end{array} \right] \&=\left[ \begin{array} { r } 5 \12 \7\end{array} \right]\end{aligned}$$

-Suppose B={b1,b2}B = \left\{ b _ { 1 } , b _ { 2 } \right\}B={b1​,b2​}

-Suppose $B = \left\{ b _ { 1 } , b _ { 2 } \right\}$