Final answer:
The standard matrix for the described transformation is a multiplication of the shear transformation matrix and the reflection matrix. Since the determinant is non-zero, the transformation is invertible, and the standard matrix for the inverse transformation is provided.
Step-by-step explanation:
To find the standard matrix of a linear transformation from ℝ² to ℝ² that first performs a vertical shear and then reflects through the x₂ axis, we begin by examining the effect on the basis vectors e₁ and e₂. A vertical shear mapping e₁ to e₁ + 3e₂, while leaving e₂ unchanged, can be represented by the matrix
A =
1 0
3 1
The reflection through the x₂ axis negates the second component of any vector, so its matrix is
B =
1 0
0 -1
The overall transformation is represented by the matrix product BA, which yields
T =
1 0
-3 -1
For invertibility, we check if the determinant of T is non-zero:
det(T) = -(1)(-1) - (0)(-3) = 1 ≠ 0
Since the determinant is non-zero, T is invertible. The inverse of T, T⁻¹, can be found by inverting matrix T:
T⁻¹ =
1 0
3 -1
This matrix T⁻¹ is the standard matrix for the inverse transformation T⁻¹.