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# Definition of identity linear transformation

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

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