Final answer:
The linear transformation L(x) = (-x₁, x₂)T is a linear operator on ℝ² and reflects a point across the y-axis.
Step-by-step explanation:
Linear transformation L(x) = (-x₁, x₂)T is a linear operator on ℝ² because it satisfies the properties of additivity and scalar multiplication. To show this, we can consider two vectors A and B in the plane. Let A = (a₁, a₂)T and B = (b₁, b₂)T. Then, applying the linear transformation, we have: L(A + B) = L((a₁ + b₁, a₂ + b₂)T) = (-(a₁ + b₁), a₂ + b₂)T = (-a₁ - b₁, a₂ + b₂)T = (-a₁, a₂)T + (-b₁, b₂)T = L(A) + L(B).
Similarly, for scalar multiplication, let k be a scalar. Then: L(kA) = L((ka₁, ka₂)T) = (-(ka₁), ka₂)T = k(-a₁, a₂)T = kL(A). Therefore, L(x) = (-x₁, x₂)T is a linear operator.
Geometrically, this linear transformation reflects a point across the y-axis. The x-coordinate of a point is negated, while the y-coordinate remains the same. For example, if we take the point A = (2, 3), applying the linear transformation L(A) results in (-2, 3).