- #1
"Don't panic!"
- 601
- 8
Hi,
I'm trying to justify to myself the abstract notion of a vector space and I would really appreciate if people wouldn't mind taking a look at my description and letting me know if it's correct, and if not, what is the correct explanation? :
"Vectors are most often introduced as ordered arrays of numbers (ordered "n-tuples") in [itex]\mathbb{R}^{n}[/itex] (or [itex] \mathbb{C}^{n}[/itex]), specifying their components along each coordinate line in Euclidean space. However, this viewpoint is very restrictive, as it requires one to introduce a specific coordinate system in order to describe a vector; an often non-trivial process, as it is not necessarily obvious which is the best coordinate system to choose. Vectors are intrinsically geometric entities, possessing both magnitude \& direction; they are mathematical objects that exist independently of any given coordinate system. Thus, we require a coordinate-free definition of a vector, and the 'space' that it lives in. We do this by abstracting the definition of a Euclidean vector in [itex]\mathbb{R}^{n}[/itex] (or [itex] \mathbb{C}^{n}[/itex]), retaining only the properties that are characteristic to the vectors themselves, and not a particular coordinate system. The characteristic properties that we refer to are the binary operations of vector addition and scalar multiplication."
Apologies in advance for the lack of mathematical rigour, I'm a physics student, but I am very interested in the mathematical formalism.
I'm trying to justify to myself the abstract notion of a vector space and I would really appreciate if people wouldn't mind taking a look at my description and letting me know if it's correct, and if not, what is the correct explanation? :
"Vectors are most often introduced as ordered arrays of numbers (ordered "n-tuples") in [itex]\mathbb{R}^{n}[/itex] (or [itex] \mathbb{C}^{n}[/itex]), specifying their components along each coordinate line in Euclidean space. However, this viewpoint is very restrictive, as it requires one to introduce a specific coordinate system in order to describe a vector; an often non-trivial process, as it is not necessarily obvious which is the best coordinate system to choose. Vectors are intrinsically geometric entities, possessing both magnitude \& direction; they are mathematical objects that exist independently of any given coordinate system. Thus, we require a coordinate-free definition of a vector, and the 'space' that it lives in. We do this by abstracting the definition of a Euclidean vector in [itex]\mathbb{R}^{n}[/itex] (or [itex] \mathbb{C}^{n}[/itex]), retaining only the properties that are characteristic to the vectors themselves, and not a particular coordinate system. The characteristic properties that we refer to are the binary operations of vector addition and scalar multiplication."
Apologies in advance for the lack of mathematical rigour, I'm a physics student, but I am very interested in the mathematical formalism.