If and and are arbitrary vectors, then prove that
This problem appeared in my Differential Geometry class, the professor explained the problem by, first taking an arbitrary vector and demonstrating that . and then proceeded to demonstrate that, . I get the proof somewhat. Can any of you elucidate it or give an alternative proof?
Answer & Explanation
are the component of the tensor product of two vector . Among all these tensors there are also the tensors where is a base of the vector space V,and is a base of the space of rank 2 tensors over tensors over V.
is the inner product of the tensor and the tensor
You know that if the inner product of a vector for each element of a base vanish, then the vector is the null vector. This is true also for the inner product vector space of tensors.
Fix, h,k, then if you take
for the arbitrariness of h,k, this is true for all h,k.
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