What is the requirement for something to qualify as vector?

[Wrichik Basu]

I have been reading Ramamurti Shankar's book "Principles of Quantum Mechanics". The author, in the first chapter, briefs out the elementary mathematics required for quantum mechanics.

Now, the author has described vector spaces, and made it very clear that only arrowed vectors that one studies in elementary physics, are not the only vectors, but matrices and functions may also be considered as vectors in a vector space consisting of such elements.

But how does something qualify as a vector? Does that quantity have to properly conform to the axioms of vectors (a picture from the book given below shows the axioms), or are there some other specific rules to which it must conform to qualify as a vector?

 

[Orodruin]

A vector is an element in a vector space. The image from your book shows the definition of a vector space.

 

[Wrichik Basu]

A vector is an element in a vector space. The image from your book shows the definition of a vector space.

Yes. It shows the definition of a vector space. But in bullet points, it also shows some axioms followed by vectors. How can I understand any quantity is a vector? Can I only determine that if a vector space is already specified?

 

[Orodruin]

As stated in the picture "A linear vector space ... is a collection of objects, called vectors". There really is nothing more to it. An object is a vector if it is an element of a vector space.

 

[Wrichik Basu]

As stated in the picture "A linear vector space ... is a collection of objects, called vectors". There really is nothing more to it. An object is a vector if it is an element of a vector space.

OK, I see. So I can't make out a vector separately without a vector space specified.

Thank you for your input.

 

[fresh_42]

For short: A vector is an object that can be added to others of its kind, as well as stretched or compressed, usually something with a direction and a length.

Of course this is very short. E.g. the factors for compression or stretching don't need to be real numbers. Sometimes it is only $\{0,1\}$ or just rational numbers. They do have in common, that they are commutative numbers and those unequal to zero can be inverted. Another point is the imagination of a vector as an arrow (direction and length). Although this is basically true, it is hard to imagine functions as arrows. However, e.g. continuous functions do form a vector space ($(f+g)(x)=f(x)+g(x)$ and $(c\cdot f)(x) = c \cdot f(x)$ fulfill the criteria), so they are elements of a vector space and thus vectors, despite the fact that we do not associate arrows with them. So the picture of arrows (e.g. forces or velocities) is often helpful, but isn't helpful in all cases.

OK, I see. So I can't make out a vector separately without a vector space specified.

You can. If you simply consider one object, which can be stretched and compressed by numerical factors (also negative ones), then you get a vector space and your object is up to length the only element, i.e. a vector. But, yes, formally it is again a vector space.

 

[FactChecker]

Yes. It shows the definition of a vector space. But in bullet points, it also shows some axioms followed by vectors. How can I understand any quantity is a vector? Can I only determine that if a vector space is already specified?

The definition may have made it confusing by called the elements vectors at the beginning. They are just symbols of elements in a set. If the elements and the set has the properties specified, then it is a vector space and the elements are vectors. I believe that the definition is still formally correct as it is, but confusing if you think that "called vectors" means anything other than the properties that follow in the definition.