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They should not be calles vectors both. Horizontal vectors are forms (maps from R^n to R). Vertical vectors are vectors (elements of R^n). Forms applied to ve
by pfortuny 4y ago
They should not be calles vectors both.
Horizontal vectors are forms (maps from R^n to R). Vertical vectors are vectors (elements of R^n).
Forms applied to vectors are “rows multiplied by columns”, and so: numbers.
Same thing as bras and kets, by the way.
- lupire 4y agoThis is a great example of how math is generally weak at types (constantly doing implicit casts and conversions with ambiguous notation, all the way back to Liebniz notation for calculus, and probably further), and the computer scientists add clarity. It's also an example of how "making it simpler" by dropping types (old Python, Bash, typeless vectors and matrices) ends up making it harder to get work done right when the ideas are non-trivial.
- pfortuny 4y agoWell, yes. But it is another way of saying that we think in analogies. At the same time, forms are also vectors… and vectors are forms of the dual… So it is not that easy.
- captainmuon 4y agoIf you have two pointy arrows, and want to know if they are perpendicular, then you use the scalar product u⋅v and check if it is 0. There is no difference between u and v in this case. You could say v is a vector, and u a function taking a vector and returning a scalar (a form, a dual vector, ...). But often that is unneccessary artifice. You can just say they are both vectors in the same vector space, which is imbued with an inner product. Declaring that some objects are forms (duals, covariant vectors, ...) is only neccessary when you want to keep track of transformation properties, for example.