[Math] Are vectors and covectors the same thing

Question

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) — because the spaces overlap. However, a physicist friend of mine (I'm a physics major, too, BTW) was trying to convince me that not only are Euclidean vectors and covectors the same thing, but ANY vector is equivalent to a covector. His reasoning is that because there is an isomorphism between them and we can convert one to the other with the metric tensor, that there's no real difference between them. My argument is that an isomorphism between mathematical objects is not enough to call two things the same. For example, there is an isomorphism between the additive groups of $\Bbb R$ and $\Bbb R^2$, but surely no one would say that $\Bbb R$ is the SAME THING as $\Bbb R^2$.

So which of us is right? Are vectors the same thing as covectors?

Answer 1

To add to all the answers above, there is a delightful example in the text "Mathematics for Physics" by Stone and Goldbart, (Appendix A.3), to clarify the difference between vectors and co-vectors, which I can't resist quoting here.

One way of driving home the distinction between $V$ and $V^*$ is to consider the space $V$ of fruit orders at a grocers. Assume that the grocer stocks only apples, oranges and pears. The elements of $V$ are then vectors such as

$x = 3 \text{ kg apples }+ 4.5 \text{ kg oranges } + 2 \text{ kg pears.} $

Take $V^*$ to be the space of possible price lists, an example element being

$f = (\$3.00/\text{kg}) \text{ apples}^* + (\$2.00/\text{kg}) \text{ oranges}^* + (\$1.50/\text{kg}) \text{ pears}^*$

The evaluation of $f$ on $x$

$f(x) = 3 \times \$3.00 + 4.5 \times \$2.00 + 2 \times \$1.50 = 21.0$

then returns the total cost of the order. You should have no difficulty in distinguishing between a price list and box of fruit!