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

My Answer:

(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!

 

[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]

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]

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 60^o 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.

 

[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)...

 

[cristo]

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)...

Well that's right if it's what you get for the tranformed unit vector (1 0)^T. Do this similarly for the other unit vector, and then you will be able to write down the transformation matrix.

 

[sara_87]

thanx Cristo! happy new year