Vector vs. covector
A covector is a linear function that takes a vector and returns a number. In coordinates, both look like lists of numbers, but when the basis changes, their coordinates change in opposite ways.
Definitions
A vector is an arrow with a length and a direction.
A vector is a quantity with a magnitude and a direction. Graphically, a vector is drawn as an arrow. In coordinates, it is written as a list of numbers.
A vector is an element of a vector space: a set with vector addition and scalar multiplication satisfying the vector space axioms. Vectors are defined by what you can do with them, not by what they look like.
Examples
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$(2, 3) \in \reals^2$
$(2, 3)$ is a point of $\reals^2$, and it is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin.
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A velocity, in meters per second
A velocity is a vector because it is an arrow in space, with units of length per time. It is not a covector, because it does not take a vector and return a number.
Non-examples
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The location of Paris
The location of Paris is a point. It is not a vector, because adding the locations of Paris and the location of another city, such as Rome, has no meaning.
A covector is a rule that takes a vector and returns a number, in a way that respects adding vectors and scaling them. In coordinates, it is written as a row of numbers, which multiplies a column vector.
Consider a vector space $V$ and its scalars. A covector on $V$ is a linear map from $V$ to its scalars; an element of the dual space $V^{\ast}$. In coordinates, covectors are often written as row vectors, which act on column vectors by matrix multiplication.
Consider a manifold $M$ and a point $x$ of $M$, with its cotangent space $T_x^{\ast} M$. A covector at $x$ is an element of $T_x^{\ast} M$. The differential $\mathrm{d}f$ of a function is a covector field (a one-form).
Examples
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The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ is a covector, because it maps a column vector $v$ to the number $v_1 + 2 v_2 + 3 v_3$.
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The temperature gradient, in degrees per meter
The temperature gradient is a covector, because it takes a displacement and returns the change in temperature, a number.
Non-examples
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A velocity, in meters per second
A velocity is a vector because it is an arrow in space, with units of length per time. It is not a covector, because it does not take a vector and return a number.
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The Euclidean norm $x \mapsto \sqrt{x_1^2 + x_2^2}$ on $\reals^2$
It is not linear: the length of $-v$ is the length of $v$, not its negative.
Examples
| Example | vector | covector | Why |
|---|---|---|---|
| A velocity, in meters per second | ✓ | ✗ | A velocity is a vector because it is an arrow in space, with units of length per time. It is not a covector, because it does not take a vector and return a number. |
| The temperature gradient, in degrees per meter | – | ✓ | The temperature gradient is a covector, because it takes a displacement and returns the change in temperature, a number. |
| The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ | – | ✓ | The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ is a covector, because it maps a column vector $v$ to the number $v_1 + 2 v_2 + 3 v_3$. |
| The Euclidean norm $x \mapsto \sqrt{x_1^2 + x_2^2}$ on $\reals^2$ | – | ✗ | The Euclidean norm takes a vector and returns a number, but it is not a covector, because it is not linear: $\lVert -x \rVert = \lVert x \rVert$. |
Why coordinates change differently
In one dimension, suppose we change units from meters to centimeters, so the new basis vector is $100$ times shorter than the old one.
- A displacement of $2$ m becomes $200$ cm. Its coordinate is multiplied by $100$: the coordinates of a vector change opposite to the basis (they are contravariant).
- A temperature gradient of $5$ degrees per meter becomes $0.05$ degrees per centimeter. Its coordinate is divided by $100$: the coordinates of a covector change with the basis (they are covariant).
The value of a covector applied to a vector does not depend on the basis: $5 \times 2 = 0.05 \times 200 = 10$ degrees.
Notes
- With an inner product, each vector $v$ determines a covector $w \mapsto \langle v, w \rangle$. This is how the gradient (a vector) is obtained from the differential (a covector), and why the two are often confused.