Consider the bases B = left(begin{array}{c}begin{bmatrix}2

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Consider the bases B = left(begin{array}{c}begin{bmatrix}2

aortiH 2020-11-14 Answered

Consider the bases $B = (\begin{matrix} [\begin{matrix} 2 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 5 \end{matrix}] \end{matrix}) o f R^{2} a n d C = (\begin{matrix} [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] \end{matrix}, [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}]) o f R^{3}$ .
and the linear maps $S \in L (R^{2}, R^{3}) a n d T \in L (R^{3}, R^{2})$ given given (with respect to the standard bases) by $[S]_{E, E} = [\begin{matrix} 2 & - 1 \\ 5 & - 3 \\ - 3 & 2 \end{matrix}] a n d [T]_{E, E} = [\begin{matrix} 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix}]$ Find each of the following coordinate representations. $(b) {[S]}_{E, C}$
$(c) {[S]}_{B, C}$

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Caren

Answered 2020-11-15 Author has 96 answers

$B = (\begin{matrix} [\begin{matrix} 2 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 5 \end{matrix}] \end{matrix}) o f R^{2}, C = (\begin{matrix} [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] \end{matrix}, [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}]) o f R^{3}$ .

$S \in L (R^{2}, R^{3}), T \in L (R^{3}, R^{2})$
$[S]_{E, E} = [\begin{matrix} 2 & - 1 \\ 5 & - 3 \\ - 3 & 2 \end{matrix}], [T]_{E, E} = [\begin{matrix} 1 & - 1 & 1 \\ 1 & 1 & - 1 \end{matrix}]$

(b) To find ${[S]}_{E, C} :$
$S ([\begin{matrix} 1 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 5 \\ - 3 \end{matrix}] = a [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}] + b [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] + c [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} a + b \\ a + c \\ b + c \end{matrix}]$
$∴ a + b = 2, a + c = 5, b + c = - 3$
$b = 2 - a \Rightarrow b + c = 2 - a + c = - 3 \Rightarrow - a + c = - 5$
$\begin{array}{cc} ∴ a + c = 5 \\ + - a + c = - 5 \\ 2 c = 0 \end{array} \begin{array}{cc} \Rightarrow c = 0, b = - 3, a = 5 \\ \Rightarrow a = 5, b = - 3, c = 0 \end{array}$
$S ([\begin{matrix} 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 1 \\ - 3 \\ 2 \end{matrix}] = a [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}] + b [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] + c [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} a + b \\ a + c \\ b + c \end{matrix}]$
$∴ a + b = - 1, a + c = - 3, b + c = 2$
$∴ b = - 1 - a \Rightarrow b + c = - 1 - a + c = 2 \Rightarrow - a + c = 3$
$\begin{array}{cc} ∴ a + c = - 3 \\ + - a + c = 3 \\ 2 c = 0 \end{array} \begin{array}{cc} \Rightarrow c = 0, a = - 3, b = 2 \\ \Rightarrow a = - 3, b = 2, c = 0 \end{array}$
$∴ [S]_{E, C} = [\begin{matrix} 5 & - 3 \\ - 3 & 2 \\ 0 & 0 \end{matrix}]$

c) To find ${[S]}_{B, C}$

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Consider the bases B = left(begin{array}{c}begin{bmatrix}2

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