Find the matrix representing the transformation

In summary, the conversation discusses finding a transformation matrix for different transformations (reflection, rotation) of the unit vectors in R2. Tips and equations are provided to help solve for the matrix representation, and specific questions are asked about certain transformations. The conversation also addresses the importance of starting from the right when constructing a transformation matrix and the use of trigonometric functions in finding the transformed unit vector. The conversation ends with a greeting and well wishes for the new year.
  • #1
sara_87
763
0
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!
 
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  • #2
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?
 
  • #3
sara_87 said:
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:

[tex] \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)[/tex]
and
[tex] \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)[/tex]

From this, we obtain four equations which we can solve for a,b,c,d.
 
  • #4
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?
 
  • #5
sara_87 said:
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 [itex] sin(-\theta)=-sin(\theta), cos(-\theta)=cos(\theta) [/itex])
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.
 
Last edited:
  • #6
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)...
 
  • #7
sara_87 said:
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.
 
  • #8
thanx Cristo! happy new year
 

What is a matrix?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is a mathematical tool used to represent linear transformations and systems of equations.

What is a transformation?

A transformation is a mathematical operation that changes the position, size, or shape of a geometric figure. It can also refer to a process that converts one set of data into another set.

How do you find the matrix representing a transformation?

To find the matrix representing a transformation, you need to identify the input and output variables of the transformation. Then, you can construct a matrix with the coefficients of the variables as its elements, which will represent the transformation in matrix form.

What do the elements of the matrix represent?

The elements of the matrix represent the coefficients of the input variables in the transformation. Each row of the matrix corresponds to a specific output variable, while each column corresponds to a specific input variable.

Why is it important to find the matrix representing a transformation?

Finding the matrix representing a transformation is important because it allows us to perform calculations and analyze the transformation using matrix operations. It also allows us to easily apply the transformation to different sets of data or equations.

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