# 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.