Ideal representation for vectors/covectors

[Jhenrique]

A vector can be expressed by a column matrix or by a row matrix, however is preferably use the column matrix for represent vectors and row matrix for covectors.

So, analogously, we can express a vector v as v₁ e¹ + v₂ e² or as v¹ e₁ + v² e₂. But I think that a vector is represented more correctly with one of those two possibilities, whereas that a covector is better represented with the other. The question is, which representation is the ideal representation for a vector and for a covector?

 

[micromass]

A vector can be expressed by a column matrix or by a row matrix, however is preferably use the column matrix for represent vectors and row matrix for covectors.

So, analogously, we can express a vector v as v₁ e¹ + v₂ e² or as v¹ e₁ + v² e₂. But I think that a vector is represented more correctly with one of those two possibilities, whereas that a covector is better represented with the other. The question is, which representation is the ideal representation for a vector and for a covector?

The notation $v_1 \mathbf{e}^1 + v_2 \mathbf{e}^2$ is used for covectors. For example, in differential geometry, you have the covectors $dx^i$.

The notation $v^1 \mathbf{e}_1 + v^2 \mathbf{e}_2$ is used for vectors. For example, tangent vectors to a manifold are given by $\frac{\partial}{\partial x^i}$ (which is seen as a lower index).

This is consistent with matrices. A matrix is written as $(\alpha^i_j)_{i,j}$, where $i$ denotes the rows and $j$ the columns. So we have, for example

$$\left(\begin{array}{cc} \alpha_1^1 & \alpha^1_2\\ \alpha^2_1 & \alpha_2^2 \end{array}\right)$$

In particular, in the case of one column, we get a vector:

$$\left(\begin{array}{c} v^1 \\ v^2\end{array}\right)$$

And in the case of one row, we get a covector:

$$\left(\begin{array}{cc} v_1 & v_2\end{array}\right)$$

Also see: http://en.wikipedia.org/wiki/Einstein_notation

 

[Jhenrique]

Thanks you!

To use the indices mixed is a thing that irritate me a lot, the indices for scalar components is the contrary of the indices of the unit vectors. I don't understand the why this...