What is the difference between a vector and a vector space?

[MNskating]

What is the difference between a vector and a vector space? I get that a vector is an object with both magnitude and direction, but am confused by the term "vector space". Does a vector space simply refer to a collection of vectors? Thanks!

 

[andresB]

I suppose you can say that a vector space is a collection of "vectors", but the issue you seems to be having is that you are using only an intuitive view of a vector. A vector space is a more abstract concept and it is not restricted to "arrows in 3d space", you should look for a book of linear algebra to learn about the subject.

 

[jtbell]

What is the difference between a vector and a vector space?

I think of it as similar to the difference between a (real) number and the number line; or the difference between a complex number and the complex (Argand) plane.

As andresB noted, the 3-d spatial vectors that we learn about in intro physics classes are only one example of the more abstract mathematical concept of "vector". The set of all 3-d spatial vectors is a vector space, but it's not the only vector space.

Knowing that some set of mathematical objects and their associated operations form a vector space is useful, because you can use your experience with other vector spaces to guide you in working with them. When I was first learning quantum mechanics as an undergraduate, it was a big "aha!" moment for me when I saw that integrals like $$\int{\psi_1^*(x)\psi_2(x)dx}$$ were like vector dot products ##\vec v_1 \cdot \vec v_2##, etc.

 

[mathman]

Basic rules of a vector space (collection of vectors). Sum of two vectors is a vector in the space. Scalar multiple of a vector is a vector in the space.

 

[bhobba]

What is the difference between a vector and a vector space? I get that a vector is an object with both magnitude and direction, but am confused by the term "vector space". Does a vector space simply refer to a collection of vectors?

You need to study linear algebra. But since this is the quantum physics sub-forum doing it using the bra-ket notation will prepare you for its use in QM:
http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/751LinearAlgebra.pdf

Thanks
Bill