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From Wikipedia:

First, we can use Einstein notation in linear algebra to distinguish easily between vectors and covectors: upper indices are used to label components (coordinates) of vectors, while lower indices are used to label components of covectors. However, vectors themselves (not their components) have lower indices, and covectors have upper indices.

I am trying to read the Wikipedia article, but I am constantly getting confused between what represents a vector/covector and what represents a component of one of these. How can I tell?

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up vote 4 down vote accepted

A vector component is always written with 1 upper index $a^i$, while a covector component is written with 1 lower index $a_i$.

In Einstein notation, if the same index variable appear in both upper and lower positions, an implicit summation is applied, i.e.

$$ a_i b^i = a_1 b^1 + a_2 b^2 + \dotsb \qquad (*) $$

Now, a vector is constructed from its component as

$$ \mathbf a = a^1 \mathbf{\hat e}_1 + a^2 \mathbf{\hat e}_2 + \dotsb $$

where $\mathbf{\hat e}_i$ are the basis vectors. But this takes the form like (*), so if we make basis vectors to take lower indices, we will get

$$ \mathbf a = a^i \mathbf{\hat e}_i $$

This is likely what Wikipedia means.

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Okay, so $a_i b^i$ doesn't represent the product of a vector times a covector, but a sum over elements. That confused me – Casebash Aug 24 '10 at 9:55
@Casebash: Right, $a_i b^i$ as a dot product is just a special case. For instance, we could use $x_i{}^i$ to represent the trace of a 2-tensor. – kennytm Aug 24 '10 at 11:13

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