What are Covectors?

Covectors are linear functions that take a vector as input and produce a scalar (a number) as output. They are described as "measuring devices" that assign a value to a vector in a way that respects linearity. For example:

If a covector measures vector \(v_1\) as \(a_1\) and \(v_2\) as \(a_2\), it measures: \(v_1 + v_2\) as \(a_1 + a_2\). If a vector \(v_1\) is scaled by a factor \(\lambda\), the covector measures \(\lambda v_1\)as \(\lambda a_1\).

Example in 2D:

A covector represented as\((a, b)\)measures a vector\((x, y)\)by computing\(ax + by\), a linear combination that yields a scalar.