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- Rizwan Ahmed
Friday, January 20, 2006
  
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Dear Rizwan

Span is nothing in Linear algebra but we have the term span of vectors or spanning set of the given vectors. Suppose that we have vectors v1,v2,v3,,vn then Span{ v1,v2,v3,,vn} is a set containing all the elements of the type c1v1 + c2v2 + c3v3 + + cnvn where c1,c2,c3,,cn are the scalars and note that span of these vectors has infinite elements. So span of given vectors is also a set which contain all the possible linear combinations of these vectors. Also remember that if we have vectors v1,v2,v3,,vn then c1v1 + c2v2 + c3v3 + + cnvn is known as linear combination of the vectors c1,c2,c3,,cn.  

Let us see What is Span of the vectors (1, 1, 0), (0, 1, 4) and (-1, 0, 2) is ?

First of all you should note that Span{(1, 1, 0), (0, 1, 4), (-1, 0, 2)}is a set containing the elements of the form c1(1, 1, 0) + c2(0, 1, 4) + c3(-1, 0, 2) where c1, c2 and c3 are scalars means these are real numbers. Now as there are infinite real numbers which can take values as c1, c2 and c3  it means that Span{(1, 1, 0), (0, 1, 4), (-1, 0, 2)}contains infinite elements. We list some of these vectors as

if we take c1= 1 ,c2= -2 and c3= 5 the we have 1(1, 1, 0) - 2(0, 1, 4) + 5 (-1, 0, 2) = (1 + 0 - 5, 1 - 2 + 0 , 0 - 8 + 10) =(-4, -1, 2 ) is in the Span{(1, 1, 0), (0, 1, 4), (-1, 0, 2)}. If we take c1= 0 ,c2= 0 and c3= 0 the clearly the vector (0, 0, 0) is also in Span{(1, 1, 0), (0, 1, 4), (-1, 0, 2)}.

We can also check whether the set of vectors spanR3 or not, and the given set of vectors will span the R3 if an arbitrary element of R3 can be written as c1(1, 1, 0) + c2(0, 1, 4) + c3(-1, 0, 2). Let (a, b, c) be any element of R3 then we will solve   

c1(1, 1, 0) + c2(0, 1, 4) + c3(-1, 0, 2)=(a, b, c) which is equivalent to the following system of linear equations.

 

                            c1           -  c3   = a

                            c1 +   c2             = b

                                   4c2 + 2c3  = c

If this system is consistent then Span{(1, 1, 0), (0, 1, 4), (-1, 0, 2)}= R3.



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What is meant by SPAN? Please write it in simple words.  
 
