Linear Transformation

• Let and be a Vector Space. A function is called a linear transformation from into if and we have

  We refer to as linear

  • If is linear, then
  • is linear if and only if for all and
  • is linear if and only if and we have

• The identity transformation is defined as where

  The zero transformation is defined as by

• A linear operator is of the form , that is, it is an endomorphism.

• (Friedberg 2.7) The space of linear transformations form a vector space. More formally, let be arbitrary functions and .

  Define and .

  Then is linear and the collection of linear transformations form a vector space denoted .

Topics §

• Dimension Theorem
• Linear Transformation Matrix Isomorphism
• Matrix Diagonalization
• Invariant Subspace
• Projection
• Bilinear Form
• Quadratic Form

Links §

• Friedberg, Insel and Spence - Ch. 2