= Covector
{title2=$\varphi\in V^*$}
A <covector> is an element of the <dual space>, hence a <linear functional> on a <vector space>. In finite-dimensional coordinates, it acts as a row multiplied by a <coordinate vector> column. The row $\begin{pmatrix}2&-1\end{pmatrix}$ acts on $\begin{pmatrix}3\\1\end{pmatrix}$ to give five. The terminology emphasizes its role as a scalar-valued measurement and its behavior under a <change of basis>.
For example, the <total derivative> of $f(x,y)=x^2+y^2$ at $(1,2)$ sends a small displacement $h$ to the linear change $df(h)=2h_x+4h_y$. This <covector> describes the first-order response. The <gradient> $\nabla f=(2,4)$ represents the same rule through the chosen Euclidean <inner product>. In <differential geometry>, the <cotangent space> collects these scalar-valued measurements of <tangent vectors>.
Back to article page
