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Definition of identity linear transformation

Created over 8 years ago, updated 10 days ago

Definition: The identity linear transformation on a vector space (V), denoted (I_V) (or simply (I)), is the map:
[I_V: V o V, \quad I_V(\mathbf{v}) = \mathbf{v} \quad ext{for all } \mathbf{v} \in V]

The identity transformation sends every vector to itself. It is the simplest linear transformation and serves as the multiplicative identity in the algebra of linear operators.

Properties:

  • (I_V) is linear: (I_V(\mathbf{u} + \mathbf{v}) = \mathbf{u} + \mathbf{v} = I_V(\mathbf{u}) + I_V(\mathbf{v})) and (I_V(c\mathbf{v}) = c\mathbf{v} = cI_V(\mathbf{v}))
  • (I_V) is invertible and (I_V^{-1} = I_V)
  • For any linear transformation (T: V o W), we have (T \circ I_V = T) and (I_W \circ T = T)
  • The matrix of (I_V) with respect to any basis is the identity matrix (I_n)