Final answer:
The linear transformation reflecting points about the x-z plane changes the y-coordinate's sign while keeping the x and z coordinates the same, which can be represented by a matrix with diagonal entries (1, -1, 1).
Step-by-step explanation:
The reflection about the x-z plane in three-dimensional space can be represented by the following linear transformation T: R^3 -> R^3:
T( x \\ y \\ z ) = [x \\ -y \\ z ]
This transformation changes the sign of the y-component while leaving the x and z components unchanged. It reflects points across the x-z plane.
In matrix form, the transformation matrix A for this reflection is:
A = [1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1]
So, the linear transformation T is represented by the matrix A, and applying this transformation to a vector in {R}^3 will give the reflection about the x-z plane.