They transform contravariantly or covariantly, respectively, with respect to change of basis.
In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its components, as in:
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see Covariance and contravariance of vectors.
The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.
As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
Using an orthogonal basis, the inner product is the sum of corresponding components multiplied together:
This can also be calculated by multiplying the covector on the vector.
Again using an orthogonal basis (in 3 dimensions) the cross product intrinsically involves summations over permutations of components: