# Find the matrix representing the transformation

#### sara_87

Hello all, can anyone help me with this question?

Question:

By considering the images of the unit vectors (1 0)T and (0 1)T in R2 under the following transformations, find the matrix representing the transformation.
(i)Reflection in the line y = -x
(ii)Reflection in the line y = mx
(iii)Rotation (anticlockwise) through 60º
(iv)Rotation (anticlockwise) through general angle theta

(i) i drew the x and y axis and i plotted (1 0)T and (0 1)T then i did the reflection of those pionts with respect to the line y=-x

(ii) STUCK

(iii) i drew the x and y axis and rotated anti clockwise by 60 degrees

(iv) STUCK

any tips would be much appreciated!

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#### sara_87

by the way for part (i) i get (0 -1) and (-1 0)...am i supossed to write it out vertically? or is my answer totally wrong?

#### cristo

Staff Emeritus
by the way for part (i) i get (0 -1) and (-1 0)...am i supossed to write it out vertically? or is my answer totally wrong?
Yea, they should be (0 -1)T and (-1 0)T.

Now, we have the reflections of the unit vectors in the line y=-x, and we wish to find the transformation matrix. We want the transformation matrix, when acting on the unit vectors in turn, to produce the transformed unit vectors.

SO, we can set up the system of equations:

$$\left(\begin{array}{cc}a&b \\c&d \end{array} \right)\left(\begin{array}{c}0\\1\end{array}\right)=\left(\begin{array}{c}-1\\0\end{array}\right)$$
and
$$\left(\begin{array}{cc}a&b \\c&d \end{array} \right)\left(\begin{array}{c}1\\0\end{array}\right)=\left(\begin{array}{c}0\\-1\end{array}\right)$$

From this, we obtain four equations which we can solve for a,b,c,d.

#### sara_87

thanx for that cristo

also for (iii) i got (1/2 root3/2) and i will put it in matrix form like you did for part (i) and as the question asked us to do
for part (iv) i think i know what to do

but for part (ii) i'm really stuck...do you know how to do it?

by the way is (1/2 root3/2) correct?

#### cristo

Staff Emeritus
thanx for that cristo

also for (iii) i got (1/2 root3/2) and i will put it in matrix form like you did for part (i) and as the question asked us to do
for part (iv) i think i know what to do

by the way is (1/2 root3/2) correct?

The unit vector (1 0)T becomes (1/2 sqrt(3)/2)T when rotated 60o anticlockwise.

but for part (ii) i'm really stuck...do you know how to do it?
Well, here's how I'd do it. First rotate the line y=mx onto the x axis, then reflect in the x axis, and finally rotate the line back. (There is a simple relation between the angle of rotation and the gradient, m). If you have done the rotation matrix correctly, this should be quite straightforward, but a few things to note.

1. The first rotation is clockwise unlike rotation in part (iv). Hence, replace theta with -theta (N.B $sin(-\theta)=-sin(\theta), cos(-\theta)=cos(\theta)$)
2. Note when "building" a transformation matrix like this, we must start from the right (just like applying functions). So, if T is the transformation, RO1 the rotation clockwise, R the reflection, and RO2 the rotation anticlockwise, then T=(RO2)R(RO1), where the products are matrix multiplication.

Post your attempts, and I'll help further if you need.

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#### sara_87

i drew the line y=mx then we get an angle theta, i reflected that then we get 2theta
so i got cos(2theta) and sin(2theta)...

Staff Emeritus