202501111439
Status: #idea
Tags: Linear Algebra, Tensor Calculus, Vectors
State: #nascient
Covectors
They are a special type of vector which live in the Dual Space which is a vector space that is tangent to the vector space of any vector space we are typically used.
The elements in that space are functions that act on vectors of the so-called Tangent Space to output a real number.
The reason behind their name of covector can be understood in too ways:
- They are subordinate to the vectors in the tangent space, they are like the shadow cast by a light. They exist because the latter exist.
- They are Covariant tensors, their components vary with their basis. They co-vary. This is why contrarily to typical vectors, the intuition of using forward to change from old to new, and backward to change from new to old is spot on. Because they do the same thing as the basis.
I must state STRONGLY, the basis of the dual space in general is NOT the basis of the tangent space. Still, the covectors' components will vary in the same way as the basis of that dual space.