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\HHT hyperplain h\\B#aaddcc

  \fvs3\bHyperplain \  \_\_

  \b Dimension 2 \\_\_

  If our vector space V is a R\]3 \ or well known from High Schoo Geometry Euclidian space E\]3 \ , then
  there is a very good association between linear subspaces in V and plains in E\]n \ 
  (As it is known, R\]n \  and vectors in E\]n \ are isomorphic, so in this page we will not distigvish them.) \_\_
  
  Indeed, if one takes two vectiors belonging one plain, then summ of these vectiors will also belong this
  plain. Multiplying these vectors with a number will also give us vector belonging to the same plain.
  Moreover, if vice versa, one will take vector from the same plain, then this vector can be expressed as
  linear combination of two given independent vectors from the same plain. Continuing speaking in the same free
  manner, one can say that plains in R\]3 \ are subspaces with dim = 2.\_\_

  Precisely speaking, \_
  any plain in R\]3 \ containing origin (0,0,0) is a subspace of  R\]3 \  with dim = 2, and\_
  any such a subspace is a plain containing this origin.\_\_

  Proof. \_\_

  Let S be a subspace of R\[2 \ and dim S = 2.\_\_
  Let u,v be a basis of S. u and v are linearly independent, so vector product h=[u x v] is not a zero.
  Any vector w from S is a linear combination of u and v, so h and w are perpendicular. Because vectors of R\[3 \ 
  have their beginning in origin O when one assossiate them with points in E\[3 \ , then one obtains that S is 
  comprised with vectors belonging plain P perpendicular to h. Using Elemental Geometry, it is easy to prove that,
  vice versa, any vector from P belongs to S, so S=P.  
  \_\_

  Let P be a plain containing origin O. We can take two \Arvector_space.asp#independent"  independent \ vectors in P, 
  and \Arbasis.asp#generators"  build \ subspace S. As this is known from Elemental Geometry, any linear combination
  of these two vectors will belong P, so S will belong P. Any vector from P aslo can be expressed via this two vectors, so
  P belongs S also.  Therefore, S = P. \_\_

  \bDimensions other than 2. \\_\_

  To recall above mentioned assosiation between plains and subspaces in our texts, we will use term 
  "hyperplain" for subspace with dimension n-1, where n is a dimension of given space.
  For example, if one has R\]2 \ and subspace dim = 1 which is simply a line crossing origin O, 
  we may say that this line is a hyperplain in R\]2 \ .

h\