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Let $T:\Bbb{R}^2\rightarrow\Bbb{R}^2$ be the linear transformation given by $T(x,y)=(x+2y,x).$

Briefly describe the geometric effect of the linear transformation $T$.

I found out the matrix representation of this transformation with respect to the standard basis of $\Bbb{R}^2$, which is $$ \begin{bmatrix} 1 & 2 \\ 1 & 0 \\ \end{bmatrix} $$ I tried plotting out some coordinates as well but I couldn't see a pattern. How can I know the geometric effect of this transformation?

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  • $\begingroup$ That is not a map from $\Bbb R$ into $\Bbb R$! $\endgroup$ – José Carlos Santos May 27 at 9:46
  • $\begingroup$ It is now. Just edited. $\endgroup$ – CJC .10 May 27 at 9:48
  • $\begingroup$ Figure out the images of lines of constant $x$ or constant $y$. $\endgroup$ – Yves Daoust May 27 at 9:48
  • $\begingroup$ You could try to consider how it behaves on the standard basis. Maybe see what happens to a unit cube? $\endgroup$ – zo0x May 27 at 9:49
  • $\begingroup$ @YvesDaoust I don't quite understand. Can you be more specific? Thank you! $\endgroup$ – CJC .10 May 27 at 9:49
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To see the geometric action of the linear transformation $$ T = \begin{bmatrix} 1 & 2 \\ 1 & 0 \end{bmatrix} $$ Consider the following basic geometric transformations:

Stretching by $ (a,b) $ $$ S = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} $$ Skewing by $ c $ (in $ x $-direction) $$ Z = \begin{bmatrix} 1 & c \\ 0 & 1 \end{bmatrix} $$ Mirroring $$ M = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$ Rotation by $ \theta $ $$ R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $$ Can you somehow determine a combination of these which will produce the same transformation?

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  • $\begingroup$ Nope, I cannot make out any combinatons... $\endgroup$ – CJC .10 May 27 at 10:43
  • $\begingroup$ Try to use SZM and determine a,b and c. $\endgroup$ – zo0x May 27 at 11:58

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