a242c8…: “Due to the super approximation”

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True/False: Let $T:V\to W$ be a linear transformation. The matrix for $T$ relative to the bases $\mathcal {B}$ and $\mathcal {C}$ for $V$ and $W$ respectively is given by: $M =\begin {bmatrix} [T(\vec {b}_1)]_{\mathcal {C}}& [T(\vec {b}_2)]_{\mathcal {C}} &\cdots & [T(\vec {b}_n)]_{\mathcal {C}}\end {bmatrix}$ where $\mathcal {B} = \{\vec {b}_1,\vec {b}_2, \dots , \vec {b}_n \}$ .

True False

True/False: Let $T:V\to W$ be a linear transformation. Let $\mathcal {B}$ and $\mathcal {C}$ be bases for $V$ and $W$ respectively. Let $M$ be the matrix for $T$ relative to $\mathcal {B}$ and $\mathcal {C}$ . Then which of the following equations is true?

$[T(\vec {x})]_\mathcal {C} = M[\vec {x}]_\mathcal {B}$ $[T(\vec {x})]_\mathcal {B} = M[\vec {x}]_\mathcal {C}$

Let $\mathcal {B} = \{\vec {b}_1, \vec {b}_2\}$ and $\mathcal {C} = \{\vec {c}_1, \vec {c}_2\}$ be bases for the vector spaces $V$ and $W$ respectively. Let $T:V\to W$ be a linear transformation. Given the equations below, find the matrix for $T$ relative to $\mathcal {B}$ and $\mathcal {C}$ . $T(\vec {b}_1) = 4\vec {c}_1 +2\vec {c}_2 \hspace {10pt} T(\vec {b}_2) = -3\vec {c}_2$

$M = \begin {bmatrix} \answer {4} & \answer {0}\\ \answer {2} &\answer {-3}\end {bmatrix}$

Let $\mathcal {B} = \{\vec {b}_1, \vec {b}_2,\vec {b}_3\}$ and $\mathcal {C} = \{\vec {c}_1, \vec {c}_2\}$ be bases for the vector spaces $V$ and $W$ respectively. Let $T:V\to W$ be a linear transformation. Given the equations below, find the matrix for $T$ relative to $\mathcal {B}$ and $\mathcal {C}$ . $T(\vec {b}_1) = -\vec {c}_1 +2\vec {c}_2 \hspace {10pt} T(\vec {b}_2) = \vec {c}_1+\vec {c}_2 \hspace {10pt} T(\vec {b}_3) = 5\vec {c}_1$

$M = \begin {bmatrix} \answer {-1} & \answer {1}& \answer {5}\\ \answer {2} &\answer {1}&\answer {0} \end {bmatrix}$

Let $\mathcal {B}$ be a basis for some vector space $V$ . If the linear transformation $T: V\to V$ sends vectors written with respect to the basis $\mathcal {B}$ to vectors written with respect to the basis $\mathcal {B}$ , then the matrix for $T$ relative to $\mathcal {B}$ (or the $\mathcal {B}$ -matrix for $T$ ) satisfies: $[T(\vec {x})]_\mathcal {B} =[T]_\mathcal {B}[\vec {x}]_\mathcal {B}$

True False

True/False. Suppose $A=PDP^{-1}$ where $D$ is a diagonal $n\times n$ matrix. If $\mathcal {B}$ is the basis for $\RR ^n$ formed fromt he columns of $P$ , then $D$ is the $\mathcal {B}$ -matrix for the transformation $\vec {x} \mapsto A\vec {x}$ .

True False

See the diagonal matrix representation theorem on page 291 of Lay.

Suppose $A = PDP^{-1}$ where $P,D, P^{-1}$ are given below. Let the linear transformation $T:\RR ^3 \to \RR ^3$ be defined by $T(\vec {x}) = A\vec {x}$ . Which of the following gives a basis $\mathcal {B}$ for $\RR ^3$ with the property that $[T]_\mathcal {B}$ is diagonal.

$P = \begin {bmatrix} 3 & 0 &-1\\ 0& 1&-3 \\1&0&0\end {bmatrix} \hspace {10pt} D = \begin {bmatrix} 3 & 0 &0\\ 0& 4&0 \\0&0&3\end {bmatrix} \hspace {10pt} P^{-1} = \begin {bmatrix} 0 & 0 &1\\ -3& 1&9 \\-1&0&3\end {bmatrix}$

$\begin {bmatrix} 3\\0\\0\end {bmatrix},\begin {bmatrix} 0\\4\\0\end {bmatrix}, \begin {bmatrix} 0\\0\\3\end {bmatrix}$ $\begin {bmatrix} 3\\-3\\0\end {bmatrix},\begin {bmatrix} 0\\4\\0\end {bmatrix}, \begin {bmatrix} 0\\9\\3\end {bmatrix}$ $\begin {bmatrix} 3\\0\\1\end {bmatrix},\begin {bmatrix} 0\\1\\0\end {bmatrix}, \begin {bmatrix} -1\\-3\\0\end {bmatrix}$

See the diagonal matrix representation theorem on page 291 of Lay.