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

  1. May 21, 2015 #1
    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!
  2. jcsd
  3. May 21, 2015 #2
    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.
  4. May 21, 2015 #3


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    Staff: Mentor

    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.
  5. May 21, 2015 #4


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    Science Advisor

    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.
  6. May 21, 2015 #5


    Staff: Mentor

    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:

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