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\HHT linear functions h\\B#aaddcc

  \fvs3\bLinear functionals \  \_\_

  In this text, we will consider finite dimensional vector spaces, and linear functions
  from V --> R. As usual, we will call such functions as linear functionals.\_\_

  Examples: \Arexample_r.asp"  V=R \ ;\Arexample_r2.asp"  V=R\]2  a\ . \-\-		
  \Arlinks_to.asp"  Links to other topics. \\_\_

  As this can follow from \Arfrom_m_to_v.asp"  more general case \ , for any given V, set F
  of all linear functionals \Arf_is_closed.asp"  is a vector space. \\_\_ 
 
  For any linear functional f, and 
  for every v from V, and every basis {e\[i \ } in V, one can express f(v) in form 
  of "linear combination" of values f on each basis vector: \_\_

  \vac
  f(v) = a\[i \ f( e\[i \ ) \-\-\-	(*) \_\_		
  v\

  Numbers a above are coordinates in formula v =  a\[i \ e\[i \ .
  \Arsummation.asp"  Summation over equal indicies a\ i is assumed in above formula; i=1,..,n; n=dim V. 
  \_\_

   
  If we will keep basis { e\[i \ } fixed, from above formulas, one can see that every 
  linear functional has "own and fixed" numbers f( e\[i \ ).  If one knows these numbers,
  then action of function f on any vector v can be calculated by using formula (*).

  Vice versa, if one takes arbitrary set on n numbers  b\[1 \ , b\[2 \ , ... and "construct"
  funtction using formula (*), then this function will be (as it is easy to check) linear.

  This makes possible using these numbers which we will denote as b\[1 \ , b\[2 \ , ...  
  as indicies for every functional f:\_\_

  \vac
  f\[b1, \ \[b2, \ \[b3, \ \[.. \  = a\[i \ b\[i \    \_\_
  v\
    
  \Arexample_r.asp"  Example V=Ra\\_
  \Arexample_r2.asp"  Example V=R\]2 \ a\\_\_

  This formula has simle geometrical meaning if to consider set of points in n+1 dimensional
  space satisfying equation (*), \Arhyperplain.asp"  this will be a hyperplain. \\_\_
  
  We can look at formula (*) from another poing of view. Let us to consider function 
  P\[i \ : v |--> a\[i \ . This function assings v's i-th coordinate to v. This funcion will be
  linear; we \Arproof_vector.asp"  can a\ rewrite (*) as \_\_

  \vac  f = b\[i \ P\[i \   \-\-(**)v\\_

  This may be not a surprise that collection of "projection" functionals 
  (P\[i \ } \Arp_independence.asp"  is linearly independent \ ,
  so this collection is a basis of F, and dimension of F is n. \_\_
  
  \Arexample_r.asp#basis"  Example V=Ra\\_
  \Arexample_r2.asp#basis"  Example V=R\]2 \ a\\_\_
  

  This implies isomorphism between F and V. Indeed, as it is known, two vector spaces with the same 
  dimensions are isomorphic. Their isomorphism can be obtained if to fix their basises and set vectors
  with equal coordinates as corresponded. In our case, we can assign to each vector  v =  a\[i \ e\[i \
  functional  f\[v \  =a\[i \ P\[i \  . \-This is interesting that for V=R\]n \ , action of f\[u \ 
  gives scalar product vu, and if |u| = 1, f\[u \ is simply a projection on direction of u.\_\_

  
  \bLinear functionals and scalar product \\_\_

  ... under construction ...

  
    
  

  
  

h\