Reversing a Linear Transformation Using an Inverse Matrix

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.

Reversing a Linear Transformation Using an Inverse Matrix

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 exists.

4. When the Inverse Exists

A matrix is invertible if and only if its determinant is non-zero. This ensures:

  • No dimension collapse.
  • The mapping is one-to-one.
  • No loss of information.
  • The original vector can be recovered uniquely.

5. Why Inverse Matrices Matter

Inverse matrices are fundamental in:

  • Undoing geometric transformations.
  • Solving systems of linear equations.
  • Coordinate transformations.
  • 3D graphics and animation.
  • Physics and robotics simulations.

Whenever a linear transformation is applied, the inverse matrix allows us to move backward through the transformation and return to the original state.

© mathematics.proofs

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