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I Is zero vector always present in any n-dimensional space?

  1. Jul 28, 2017 #1
    In the book, Introduction to Linear Algebra, Gilbert Strang says that every time we see a space of vectors, the zero vector will be included in it.

    I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?
     
  2. jcsd
  3. Jul 28, 2017 #2

    andrewkirk

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    A plane in 3D space is not a vector space unless it passes through the origin. It is an Affine Space, which is a generalisation of the concept of a vector space.

    The reason that a plane that doesn't pass through the origin is not a vector space is that it is not closed under addition or scalar multiplication. Consider the plane ##z=2##, which contains the vectors (0,0,2) and (1,1,2). The sum of those is (1,1,4) which is not in that plane.
     
  4. Jul 29, 2017 #3
    Can you kindly explain that how (1,1,2) lies in the plane z=2. The plane z=2 should only contain (0,0,2). No?
     
  5. Jul 29, 2017 #4

    andrewkirk

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    No. The plane z=2 is the set of all points (x,y,2) where x and y are any real numbers.
     
  6. Jul 29, 2017 #5
    If it would contain only one point it wouldn't be a plane...
     
  7. Jul 29, 2017 #6

    Mark44

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    To expand on what andrewkirk said, the equation z = 2 means that both x and y are completely arbitrary. You are assuming that since x and y aren't mentioned, both must be zero. Instead, since they aren't present, they can take on any values.
     
  8. Jul 31, 2017 #7
    I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?
     
  9. Aug 1, 2017 #8
    If the plane does not pass through the origin then it is not a vector space (according to definition a vector space must contain the zero vector).
     
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