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We just started the topic of linear transformations and I have this hw question that I just don't understand.

Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, which maps the entire Cartesian plane to the line $L = \{(x,y)\mid x = y\}$

I read up on some of the other linear transformation questions on this website but they were pretty specific questions. Could someone give me the general idea on how to solve a question asking for the existence of a linear transformation and how to prove it? Also an explanation on how to approach this problem would be much appreciated.

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    $\begingroup$ If you just take the perpendicular projection of the point $(x,y)$ in the plane on the line passing through origin at angle of $45$ degrees, that will be it. Try to figure out, how can you write in form of transformation algebraically $\endgroup$ – Bhaskar Vashishth Oct 20 '14 at 20:16
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You can always define a linear transformation by defining what it does to a basis. In your case $(1,0)$ and $(0,1)$ are a basis for $\mathbb{R}^2$. Just define a linear transformation by deciding where on the line $y=x$ they should go.

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  • $\begingroup$ Just so I understand. Do you mean manipulating the basis matrix so that when you do the transformation it maps to y = x? $\endgroup$ – user3538161 Oct 20 '14 at 20:45
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    $\begingroup$ Just send $(1,0)$ to some vector on the line $y=x$ and do the same for $(0,1)$. You can pick any two vectors on $y=x$ that you'd like. The corresponding matrix will have as first column the vector to which you sent $(1,0)$ and as second column the vector to which you sent $(0,1)$. $\endgroup$ – Joe Johnson 126 Oct 20 '14 at 23:53

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