01c5e3…: “Aside from a maroon approximation”

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True/False: If $\mathcal {B}$ and $\mathcal {C}$ are both bases of the vector space $V$ , then the $n\times n$ matrix $\underset {\mathcal {C}\leftarrow \mathcal {B}}{P}$ sends vectors in $\mathcal {B}$ -coordinates to the corresponding vectors in $\mathcal {C}$ -coordinates.

True False

Consider two bases $\mathcal {B} =\{\vec {b}_1,\vec {b}_2\}$ and $\mathcal {C} =\{\vec {c}_1,\vec {c}_2\}$ for a vector space $V$ . Suppose that $\vec {b}_1 = 3\vec {c}_1+2\vec {c}_2$ and $\vec {b}_2 = -\vec {c}_2$ .

(a): What is the $2\times 2$ change-of-coordinates matrix $\underset {\mathcal {C}\leftarrow \mathcal {B}}{P}$ ?

$\underset {\mathcal {C}\leftarrow \mathcal {B}}{P}= \begin {bmatrix} \answer {3} &\answer {0}\\ \answer {2} &\answer {-1}\end {bmatrix}$
(b): If $[\vec {x}]_{\mathcal {B}} = \begin {bmatrix}3\\-4\end {bmatrix}$ is a vector in $\mathcal {B}$ -coordinates, what is the corresponding vector in $\mathcal {C}$ -coordinates?

$[\vec {x}]_{\mathcal {C}} = \begin {bmatrix}\answer {9}\\\answer {10}\end {bmatrix}$
(c): What is the $2\times 2$ change-of-coordinates matrix $\underset {\mathcal {B}\leftarrow \mathcal {C}}{P}$ ?

$\underset {\mathcal {B}\leftarrow \mathcal {C}}{P} = \left (\underset {\mathcal {C}\leftarrow \mathcal {B}}{P}\right )^{-1}$

$\underset {\mathcal {B}\leftarrow \mathcal {C}}{P} = \frac {1}{\answer {-3}}\begin {bmatrix} \answer {-1} &\answer {0}\\ \answer {-2} &\answer {3}\end {bmatrix}$

Consider two bases $\mathcal {B} =\{\vec {b}_1,\vec {b}_2\}$ and $\mathcal {C} =\{\vec {c}_1,\vec {c}_2\}$ for a vector space $V$ . Suppose that $\vec {c}_1 = 2\vec {b}_1-2\vec {b}_2$ and $\vec {c}_2 = \vec {b}_1+3\vec {b}_2$ . Which change-of-coordinates matrix is $\begin {bmatrix} 2&1\\ -2&3 \end {bmatrix}$ ?

$\underset {\mathcal {C}\leftarrow \mathcal {B}}{P}$ $\underset {\mathcal {B}\leftarrow \mathcal {C}}{P}$

Let $\mathcal {B} =\{\vec {b}_1,\vec {b}_2\}$ and $\mathcal {C} =\{\vec {c}_1,\vec {c}_2\}$ be bases for $\RR ^2$ . Find both change-in-coordinates matrices. $\vec {b}_1 = \begin {bmatrix} 2 \\1\end {bmatrix}, \hspace {10pt} \vec {b}_2 = \begin {bmatrix}0 \\4\end {bmatrix}, \hspace {20pt} \vec {c}_1 = \begin {bmatrix} -1 \\1\end {bmatrix}, \hspace {10pt} \vec {c}_2 = \begin {bmatrix} 3 \\2\end {bmatrix}$

$\underset {\mathcal {C}\leftarrow \mathcal {B}}{P}=\begin {bmatrix} -\frac {\answer {1}}{5} &\frac {\answer {12}}{\answer {5}} \\ \frac {\answer {3}}{\answer {5}} & \frac {\answer {4}}{\answer {5}} \end {bmatrix}$

$\underset {\mathcal {B}\leftarrow \mathcal {C}}{P}\begin {bmatrix} -\frac {\answer {1}}{2} &\frac {\answer {3}}{2} \\ \frac {\answer {3}}{\answer {8}} & \frac {\answer {1}}{8} \end {bmatrix}$

$\begin {bmatrix}\vec {c}_1& \vec {c}_2 &\vec {b}_1 &\vec {b}_2\end {bmatrix} \sim \begin {bmatrix} I & \underset {\mathcal {C}\leftarrow \mathcal {B}}{P} \end {bmatrix}$