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Let V be a Vectorspace over field F spanned by a finite set fo vectors v1,v2,....,vm.Then any
Lineraly Independent subset of V is finite and contains a maximum of m elements
Solution
Let S be any arbitrary subset of V with n elements wi.
We need to show that n<=m
Enough to Show: S is LD over F such that ai belongs to F where all ai's not 0 and Summation of
ai*wi = 0
Now as wi belongs to F,we can write wi = summation(aji*vj)
Now,there exists n elements xi such that Summation(xi*wn) = 0
Substituting wi value in this equation we get
Double Summation((xi*aji)*vj) = 0
It has m equations and n unknowns
To solve we need no. of equations >= no. of variables
hence n<=m
Let V be a Vectorspace over field F spanned by a finite set fo vectors.docx

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Let V be a Vectorspace over field F spanned by a finite set fo vectors.docx

  • 1. Let V be a Vectorspace over field F spanned by a finite set fo vectors v1,v2,....,vm.Then any Lineraly Independent subset of V is finite and contains a maximum of m elements Solution Let S be any arbitrary subset of V with n elements wi. We need to show that n<=m Enough to Show: S is LD over F such that ai belongs to F where all ai's not 0 and Summation of ai*wi = 0 Now as wi belongs to F,we can write wi = summation(aji*vj) Now,there exists n elements xi such that Summation(xi*wn) = 0 Substituting wi value in this equation we get Double Summation((xi*aji)*vj) = 0 It has m equations and n unknowns To solve we need no. of equations >= no. of variables hence n<=m