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