Question

Suppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed by a reflection over the x-axis. Find the matrix A that induces T.Suppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed by a reflection over the x-axis. Find the matrix A that induces T.Suppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed by a reflection over the x-axis. Find the matrix A that induces T

Answer 1

A = \begin{pmatrix}-1 & 0 \\0 & -1\end{pmatrix}

Explanation:

Let, there be a point (x , y) in R2

After reflection on Y axis, it will be ( -x , y) and thereafter if we reflect the image over X -axis , it will be (-x , -y)

So,

we need to find out a matrix

[tex]\begin{pmatrix}a & b \\c & d\end{pmatrix}[tex]

such that,

(x , y) [tex]\begin{pmatrix}a & b \\c & d\end{pmatrix}[tex] = (-x. -y)

So, we have the following set of equations.

ax + cy = -x ------------(1)

bx + dy= -y-----------(2)

We can get the solution (by inspection) as,

[tex]\begin{pmatrix}a & b \\c & d\end{pmatrix}[tex]

= [tex]\begin{pmatrix}-1 & 0 \\0 & -1\end{pmatrix}[tex]

So, the matrix is,

A = \begin{pmatrix}-1 & 0 \\0 & -1\end{pmatrix} (answer)