Advanced Linear Algebra - Lecture 13: Properties of Linear Transformations

Introduction

• Focus on properties of linear transformations using their standard matrices.
• Key topic: Invertibility of linear transformations.

Invertibility of Linear Transformations

• Definition: A linear transformation ( t ) is invertible if there exists another linear transformation ( t^{-1} ) such that:
  • ( t^{-1}(t(v)) = v ) for all ( v ) in the input space.
  • ( t(t^{-1}(w)) = w ) for all ( w ) in the output space.
  • Composition: ( t^{-1} \circ t ) and ( t \circ t^{-1} ) both yield the identity transformation, but on different vector spaces.

Theorem on Invertibility

• Statement: A linear transformation is invertible if and only if its standard matrix is invertible.
• Note: Choice of bases influences the standard matrix, but not its invertibility.
• Proof Outline:
  1. If direction: Assume ( t ) is invertible, show standard matrix is invertible.
     • Multiply the standard matrix by its proposed inverse and check for identity.
     • Use composition properties of standard matrices.
  2. Only-if direction: If the standard matrix is invertible, then so is ( t ).
     • Reverse the logic used in the first direction.

Example: Calculating Indefinite Integrals

• Solve using standard matrices and the inverse.
• Problem: Compute ( \int x^2 e^{3x} , dx ).
  • Strategy: Use derivative standard matrix and its inverse (integration).
  • Steps:
    1. Select a basis: ( {e^{3x}, xe^{3x}, x^2e^{3x}} ).
    2. Construct the derivative standard matrix.
    3. Compute its inverse for integration.
    4. Multiply the inverse matrix by the coordinate vector of the function.
    5. Add constant ( +C ) for indefinite integral.

Properties of Invertibility

• All properties of matrix invertibility translate to finite-dimensional linear transformations:
  • Determinant non-zero.
  • Linearly independent columns.
  • Unique solution to ( ax = 0 ).
• Special Note: For finite-dimensional vector spaces of the same dimension, ( t ) is invertible if and only if ( t(v) = 0 ) implies ( v = 0 ).

Conclusion

• We can freely use matrix properties for linear transformations.
• Upcoming lectures will further explore these applications.