6d7cd2…: “Prior to the lonely lake”

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Let $A = \begin {bmatrix} 3 &0\\ 0&1 \end {bmatrix}$ and define $T: \RR ^2 \to \RR ^2$ by $T(\bx ) = A\bx$ . Let $\bu = \begin {bmatrix} 1\\-1 \end {bmatrix}$ and compute $T(\bu )$ .

$T(\bu ) = \left [ \begin {array}{c} \answer {3} \\ \answer {-1} \end {array} \right ]$

Let $A = \begin {bmatrix} -1 &2\\ 4&1 \end {bmatrix}$ and define $T: \RR ^2 \to \RR ^2$ by $T(\bx ) = A\bx$ . Let $\bu = \begin {bmatrix} -3\\1 \end {bmatrix}$ and compute $T(\bu )$ .

$T(\bu ) = \left [ \begin {array}{c} \answer {5} \\ \answer {-11} \end {array} \right ]$

Let $A$ be a $7\times 6$ matrix. What must $a$ and $b$ be in order to define $T: \RR ^a \to \RR ^b$ by $T(\bx ) = A\bx$ ?

$a= \answer {6}$ $b = \answer {7}$

Let $T:\RR ^3 \to \RR$ be a linear transformation. Then

$T({\bf 0}) = \answer {0}$