Find the matrix which represents(a)a reflection(b)a half turn abt 0(c)a reflection in the line y=x).plz help me...)
To find the matrix representation for each of these transformations, we can start with the standard basis vectors and apply the given transformation to each of them.
a) Reflection about the x-axis:
A reflection about the x-axis will change the sign of the y-coordinate while leaving the x-coordinate unchanged.
Let's consider the standard basis vectors in 2D:
i = [1 0]
j = [0 1]
To find the matrix representation of the reflection about the x-axis, we apply the reflection to each of these basis vectors:
Reflection of i about the x-axis: [1 0] -> [1 0]
Reflection of j about the x-axis: [0 1] -> [0 -1]
So, the matrix representation of the reflection about the x-axis is:
| 1 0 |
| 0 -1 |
b) Half turn about the origin:
A half turn about the origin will flip both the x and y coordinates, but also change their signs. In other words, (x, y) -> (-x, -y).
Again, let's consider the standard basis vectors,
i = [1 0]
j = [0 1]
Now we can apply the half turn to these basis vectors:
Half turn of i about the origin: [1 0] -> [-1 0]
Half turn of j about the origin: [0 1] -> [0 -1]
So, the matrix representation of the half turn about the origin is:
| -1 0 |
| 0 -1 |
c) Reflection in the line y = x:
A reflection in the line y = x can be done by swapping the x and y coordinates.
Again, let's consider the standard basis vectors,
i = [1 0]
j = [0 1]
Now we can apply the reflection to these basis vectors:
Reflection of i in the line y = x: [1 0] -> [0 1]
Reflection of j in the line y = x: [0 1] -> [1 0]
So, the matrix representation of the reflection in the line y = x is:
| 0 1 |
| 1 0 |
I hope this helps! Let me know if you have any further questions.