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Linear algebra- what is a vector space?

  1. Sep 18, 2011 #1
    I looked up what is a vector space online and it always gives like formula or long explanations. In a couple sentences can you tell me exactly what a vector space is? I know it has to go through the origin, but what else is true about a vector space?
     
  2. jcsd
  3. Sep 18, 2011 #2

    fluidistic

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    Gold Member

    I suggest you to look at http://mathworld.wolfram.com/VectorSpace.html.
    I don't know what you mean by "it has to go through the origin". A vector space must contain an element (called vector) that is neutral under addition. In other words, there must exist a vector say [itex]\vec 0[/itex] such that for any vector [itex]\vec x[/itex] in the vector space, the following hold: [itex]\vec x + \vec 0 = \vec x[/itex].
     
  4. Sep 18, 2011 #3
    It just means the 0 vector must be in that set (from memory...)
     
  5. Sep 18, 2011 #4
    If the following axioms are true for all objects u, v, and w in V and all scalars c and k then V is called a vector space and the objects in V are called vectors.

    (a) u + v is in V
    (b) cu is in V
    (c) u + v = v + u
    (d) u + (v + w) = (u + v) + w
    (e) 0 in V, such that for all u in V we have u + 0 = 0 + u = u.
    (f) For every u in V there is -u such that u + (-u) = 0.
    (g) c(u + v) = cu + cv
    (h) (c + k)u = cu + ku
    (i) c(ku) = (ck)u
    (j) 1u = u
     
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