# Thread: Describing a transformation with a matrix

1. ## Describing a transformation with a matrix

Hello

I am working on a question from a textbook about a matrix transformation. Here is the question:

I have completed the question but I disagree with the answer given in the back of the book for part (b).
I have worked out that the transformation represents a reflection in the line y=5^(-0.5) x followed by an enlargement scale factor 3.

However the book says it is a reflection in the line y=x tan 48.2 followed by an enlargement scale factor 3.

I have double checked my work and am sure I am correct. Could someone please look at my working and advise who is correct? Many thanks for looking at this.

Here is my working:

2. ## Re: Describing a transformation with a matrix

I agree with your solution ...

Started with the vector $\langle 0,a \rangle$, $a > 0$

$\begin{bmatrix} 2 & \sqrt{5}\\ \sqrt{5}& -2 \end{bmatrix} \cdot \begin{bmatrix} 0\\ a \end{bmatrix}= \begin{bmatrix} \sqrt{5} \cdot a\\ -2a \end{bmatrix}$

$\bigg|\langle 0,a \rangle \bigg|=a$ and $\bigg|\langle \sqrt{5} \cdot a,-2a \rangle \bigg|=3a$

$\langle 0,a \rangle$ has direction $\theta = 90^\circ$

$\langle \sqrt{5} \cdot a,-2a \rangle$ has direction $\phi = \arctan\left(-\dfrac{2}{\sqrt{5}}\right) \approx -41.8^\circ$

$\dfrac{\theta+\phi}{2} = \arctan\left(\dfrac{1}{\sqrt{5}}\right) \implies m = \dfrac{1}{\sqrt{5}} \text{ for } y=mx$

3. ## Re: Describing a transformation with a matrix

Thank you very much. I thought the book was wrong. Thanks for clearing this up!