Find the matrix of the linear transformation $T\colon {\Bbb R}^3 \to {\Bbb R}^2$ such that
$T(1,1,1) = (1,1)$, $T(1,2,3) = (1,2)$, $T(1,2,4) = (1,4)$.
So far, I have only dealt with transformations in the same R. Any help?
$\endgroup$ 13 Answers
$\begingroup$Group your 3 constraints into a single one:
$$\tag{1}T.\underbrace{\begin{pmatrix}1&1&1\\1&2&2\\1&3&4\end{pmatrix}}_{M}=\underbrace{\begin{pmatrix}1&1&1\\1&2&4\end{pmatrix}}_{N}$$
(where the point means matrix product).
(1) is equivalent to $T=N.M^{-1},$ which is a $2 \times 3$ matrix.
Up to you for the last calculations.
You should find $\begin{pmatrix}1& \ \ 0&0\\2&-3&2\end{pmatrix}.$
$\endgroup$ 3 $\begingroup$Hint : $T(0,0,1) = (0,2)$ so the last column of the matrix is $\begin{pmatrix} 0 \\ 2 \end{pmatrix}$.
Do you see how to find it ? Do you see how to continue ?
$\endgroup$ $\begingroup$Hint: Try finding $a$, $b$, $c$, $d$, $e$, $f$ such that: $$ \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1 \end{array} \right) = \left( \begin{array}{c} 1\\ 1 \end{array} \right) $$ What other matrix equations can you form?
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