I know the definition of a linear transformation, but I am not sure how to turn this word problem into a matrix to solve:
$T(x_1, x_2) = (x_1-4x_2, 2x_1+x_2, x_1+2x_2)$
Find the image of the line that passes through the origin and point $(1, -1)$.
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2 Answers
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The line passing through the origin and $(1, -1)$ is the set of points of the form $(t, -t)$ with $t\in\mathbb R$ (I suppose you are implicitly working over the reals). We compute $$T(t,-t)=(t-4t,2t+t,t+2t)=(-3t,3t,3t).$$ That describes the line in 3D space through the origin and $(3, -3, -3)$ (or equivalently one can use $(1, -1, -1)$ as second point)
answered Sep 12, 2012 at 5:53
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$\begingroup$ Where did you get -3t, 3t, and 3t from? $\endgroup$– Grace CCommented Sep 12, 2012 at 6:40
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$\begingroup$ I think my study guide is looking for a more specific answer... $\endgroup$– Grace CCommented Sep 12, 2012 at 7:13
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$\begingroup$ Rather, I don't see why you combine $x_1$ and $x_2$ into t. $\endgroup$– Grace CCommented Sep 12, 2012 at 7:17
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$\begingroup$ @LearningPython: Hagen wasn’t doing anything at all with $x_1$ and $x_2$ when he introduced $t$: he was describing the line through the origin and the point $\langle 1,-1\rangle$. Then he described what $T$ does to the points on that line: it sends them to the set of points of the form $s\langle 1,-1,-1\rangle$, where $s$ can be any real number. $\endgroup$ Commented Sep 12, 2012 at 7:23
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HINT: A linear transformation sends straight lines to straight lines. If $T$ sends the origin and the point $\langle 1,-1\rangle$ to the points $P$ and $Q$, it must send the line through the origin and the point $\langle 1,-1\rangle$ to the line through $P$ and $Q$.
answered Sep 12, 2012 at 5:50
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$\begingroup$ What about the use of a matrix? How would I set this up? $\endgroup$– Grace CCommented Sep 12, 2012 at 7:12
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$\begingroup$ @LearningPython: I wouldn’t use matrices for this problem: they just get in the way. The matrix representing $T$ is $\begin{bmatrix}1&-4\\2&1\\1&2\end{bmatrix}$, but having it doesn’t make anything easier. $\endgroup$ Commented Sep 12, 2012 at 7:20
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$\begingroup$ Ok, would you say that Hagen is correct? $\endgroup$– Grace CCommented Sep 12, 2012 at 7:22
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$\begingroup$ @LearningPython: Sure: he just did in more detail what I suggested you do in my hint. $\endgroup$ Commented Sep 12, 2012 at 7:24