Question

Assume that T is a linear transformation. Find the standard matrix of T. ​T: set of real numbers R squaredℝ2 first reflects points through the line x 2 equals negative x 1line x2=−x1 and then reflects points through the line x 2 equals x 1line x2=x1.

Answer 1

Answer:

$\left[\begin{array}{ccc}0&-1\\-1&-2\\\end{array}\right]$

Step-by-step explanation:

Thinking process:

We will need to test the different geometric transformations.

Let's see how this can be done:

Let the shear transformation be represented by the matrix S such that $S(e_{2}) = e_{2} + 2e_{1}$

Then, let the image be reflected by the reflection R, such that:

R = $\left[\begin{array}{ccc}1\\0\\\end{array}\right]$ is reflected across the point $x_{1} = x_{2}$

then the vector will be = $\left[\begin{array}{ccc}-1\\0\\\end{array}\right]$

This is the mirror image.

Then it means that $R (e_{1}) = -e_{2}$ and $R (e_{2}) = -e_{1}$

Thus, the standard matrix is given by: $\left[\begin{array}{ccc}0&-1\\-1&-2\\\end{array}\right]$