# Geometric effect of linear transformation

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?

• That is not a map from $\Bbb R$ into $\Bbb R$! – José Carlos Santos May 27 '20 at 9:46
• It is now. Just edited. – CJC .10 May 27 '20 at 9:48
• Figure out the images of lines of constant $x$ or constant $y$. – Yves Daoust May 27 '20 at 9:48
• You could try to consider how it behaves on the standard basis. Maybe see what happens to a unit cube? – zo0x May 27 '20 at 9:49
• @YvesDaoust I don't quite understand. Can you be more specific? Thank you! – CJC .10 May 27 '20 at 9:49

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?