Suppose that an object with two indices has the property that is a scalar for any arbitrary tensor . Show that is a tensor.
Hint: start with the relationship
Suppose that with respect to coordinate system we have that , where is a scalar. Now, in coordinate system we have that
Since are the components of a contravariant tensor of order two, we have that
Since this holds for arbitrary components , we have that
Multiplying both sides by we get
That is, are the components of a covariant tensor of order two.