Ideal representation for vectors/covectors

In summary, the ideal representation for a vector is using a column matrix, while a row matrix is preferred for representing covectors. This is consistent with the notation used for matrices, where a vector is written as a one-column matrix and a covector as a one-row matrix. The indices used for scalar components are the opposite of those used for unit vectors, which can be confusing. This is known as Einstein notation.
  • #1
Jhenrique
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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?
 
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  • #2
Jhenrique said:
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

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

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

[tex]\left(\begin{array}{c} v^1 \\ v^2\end{array}\right)[/tex]

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

[tex]\left(\begin{array}{cc} v_1 & v_2\end{array}\right)[/tex]

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

Related to Ideal representation for vectors/covectors

1. What is the difference between vectors and covectors?

Vectors and covectors are both mathematical objects used to represent quantities in physics and mathematics. Vectors are typically denoted by arrows and have both magnitude and direction, while covectors are typically represented by rows of numbers and have no direction. Vectors describe the movement or displacement of an object, while covectors describe the forces acting on an object.

2. How do you represent a vector/covector in mathematics?

In mathematics, vectors are often represented as a column matrix with numbers representing the magnitude and direction of the vector. Covectors are represented as a row matrix with numbers representing the components of the covector. Both vectors and covectors can also be represented using coordinates in a coordinate system.

3. What is an ideal representation for vectors/covectors?

An ideal representation for vectors and covectors is a notation that allows for easy manipulation and calculation of these mathematical objects. This can include using matrices, coordinates, or abstract notation such as tensor notation. The ideal representation will depend on the specific application and context in which the vectors and covectors are being used.

4. Can vectors and covectors be represented in different coordinate systems?

Yes, vectors and covectors can be represented in different coordinate systems. This is because the coordinates used to represent these objects are relative to the coordinate system being used. For example, a vector that is pointing in a certain direction in one coordinate system may have different coordinates when represented in a different coordinate system.

5. How are vectors and covectors used in physics?

Vectors and covectors are used extensively in physics to represent physical quantities such as force, velocity, and acceleration. They are also used in mathematical equations to describe physical phenomena, such as in Newton's laws of motion. Vectors and covectors are essential tools for analyzing and solving problems in physics.

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