# Proof of a Tensor's Invariance Property

The components of a type ( 0,2 ) tensor, $$T\_{a b}$$, are defined by how they transform under a change of coordinates. The proof demonstrates that if the expression $$T\_{a b} v^a w^b$$ is a scalar (meaning it remains unchanged during a coordinate transformation), then the components $$T\_{a b}$$ must transform in a specific way. This transformation rule, derived from the invariance of the scalar and the known transformation laws for vectors, is the defining characteristic of a type ( 0,2 ) tensor. Essentially, the behavior of the whole (the scalar product) dictates the behavior of the parts ( $$T\_{a b}$$ ), proving that $$T\_{a b}$$ are indeed the components of such a tensor.

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