Final answer:
The standard matrix for a linear transformation that rotates points through π radians and then reflects them across the horizontal axis is represented by multiplication of a rotation.
Step-by-step explanation:
You are asking how to find the standard matrix for a linear transformation that first rotates points through π radians (180 degrees) and then reflects points through the horizontal axis. To solve this, we need to combine the effects of both transformations. A rotation by π radians can be represented by the matrix: R = [ -1 0 0 -1]
This matrix flips the x and y coordinates since rotating 180 degrees essentially reverses the signs of both the x and y coordinates. A reflection across the horizontal axis changes the sign of the y-coordinate but leaves the x-coordinate unchanged, and is represented by the matrix: M = [ 1 0 0 -1]
Therefore, the standard matrix for the transformation T would be the product of the rotation matrix R and reflection matrix M: T = MR = [ 1 0 0 -1] [ -1 0 0 -1] =[ -1 0 0 1]. The resulting matrix T represents a rotation followed by a reflection, thus the standard matrix is: T = [ -1 0 0 1]