Geometric Bites

Reversing a Linear Transformation Using an Inverse Matrix In linear algebra, any invertible linear transformation can be reversed. The key tool that makes this possible is the inverse matrix . If a matrix transforms a vector into another, the inverse matrix recovers the original. 1. The Transformation Equation Suppose a vector x₁ is transformed into a vector x₂ using a matrix T : T x₁ = x₂ This equation describes how x₁ is mapped to x₂ . To reverse the transformation, we must apply the inverse matrix. 2. Applying the Inverse Matrix Multiply both sides of the equation by T⁻¹ : T⁻¹ (T x₁) = T⁻¹ x₂ Using the fundamental identity: T⁻¹ T = I the expression simplifies directly to: x₁ = T⁻¹ x₂ 3. Interpretation This tells us that the original vector is obtained by applying the inverse matrix to the transformed vector: Original vector = Inverse matrix × Image vector As long as the matrix is invertible, the reverse transformation always exist...