Question

Let T : R 3 → R 3 denote the reflection through the plane 3x +

3y + 6z = 0.Find the matrix A representing T in the standard
basis.

Answer 1

To find the matrix A representing the reflection T in the standard basis, we need to determine how T transforms each basis vector (i, j, k) in R^3. The matrix A representing T in the standard basis is [[-1, 0, 0], [0, -1, 0], [0, 0, -1]].

Step-by-step explanation:

To find the matrix A representing the reflection T in the standard basis, we need to determine how T transforms each basis vector (i, j, k) in R^3.

We can start by considering how T reflects the standard basis vectors.

Let's take the first standard basis vector i = (1, 0, 0). T reflects this vector through the plane 3x + 3y + 6z = 0, so the image of i under T will be i reflected through the plane. The reflection of i will be -i since it is flipped across the plane. Therefore, the first column of matrix A will be (-1, 0, 0).

We can apply the same logic to the other two standard basis vectors to find the remaining columns of matrix A.

The second column will be (0, -1, 0) and the third column will be (0, 0, -1).

Therefore, the matrix A representing T in the standard basis is:

[[-1, 0, 0], [0, -1, 0], [0, 0, -1]].