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Show that the transformation T defined by T($x_1, x_2$) = ($x_1 ^2 - 2x_2, x_1 + 5x_2$) is not linear.

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Although I answered this, I also downvoted, as I found that "this does not show any effort" (similar to the tooltip for the downvote arrow). In the future, I encourage you to put some effort into your homework, first on your own, then in your presentation of it to us. In this particular case, if you know the definition of linear transformations and try any explicit example, you'll see it's not linear. This leads one to believe that you haven't really tried much of anything. In addition, please don't write your questions in the imperative, or by directly copying the question statements. –  mixedmath Sep 10 '12 at 5:14
Ah, alright. Thank you for the advice! –  Faeynrir Sep 10 '12 at 5:27
@mixedmath I am puzzled: answering questions which show no effort is the surest way I can think of that the proportion of such questions on the site increases, irrespectively of any post hoc comment one might add to rationalize the fact of posting such answers. Very odd. –  Did Sep 10 '12 at 6:53
@did: I would argue that my answer doesn't actually answer the question content-wise at all. I would consider it very confusing if someone were to provide a complete answer and comment as I did. –  mixedmath Sep 10 '12 at 16:03
@mixedmath This does not seem the best incentive to reach acceptable questions. –  Did Sep 10 '12 at 17:00
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closed as not a real question by Andres Caicedo, William, Did, Aang, J. M. Sep 25 '12 at 12:57

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2 Answers


  1. What does it mean for a transformation to be linear?
  2. Try pretty much any trio of example computations, and see if it's linear (i.e. find a counterexample).
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In order to be linear, $T$ must satisfy the following conditions: for every $\langle x_1,x_2\rangle,\langle y_1,y_2\rangle\in\Bbb R^2$ and every $r\in\Bbb R$, $T(r\langle x_1,x_2\rangle)=r\langle x_1,x_2\rangle$, and $T(\langle x_1,x_2\rangle+\langle y_1,y_2\rangle)=T(\langle x_1,x_2\rangle)+T(\langle y_1,y_2\rangle)$. Since $r\langle x_1,x_2\rangle)=\langle rx_1,rx_2\rangle$ and $\langle x_1,x_2\rangle+\langle y_1,y_2\rangle=\langle x_1+y_1,x_2+y_2\rangle$, this translates to

  1. $T(rx_1,rx_2)=rT(x_1,x_2)$, and
  2. $T(x_1+y_1,x_2+y_2)=T(x_1,x_2)+T(y_1,y_2)$.

Now write out algebraically what $T(rx_1,rx_2)$ and $rT(x_1,x_2)$ are, and see whether they are equal for all possible choices of $r,x_1,x_2\in\Bbb R$; if they’re not, then $T$ cannot be linear. If they are, test the second condition in the same way.

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