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Linearly Dependent

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Linearly-Dependent

Linearly Dependent  Definition:

           Let V(F) be a vector space. A finite subset \(\left \{ {\alpha _{1},\alpha _{2}----\alpha _{n}}  \right \}\) of vectors of V is said to be a Linearly Dependent (L.D) set if there exist scalars\({a _{1},a _{2}----a _{n}\epsilon }  \) F, not all zero, such that

  \({a _{1}\alpha _{1}+a _{2}\alpha _{2}+----+a _{n}\alpha _{n} = \bar{0}}  \)


Probability Introduction :

Q1) Let (V,+,.) be a vector space with u,v,w \(\epsilon V\) such that \(u\neq v\) . Let w=3u+2v . Prove that {u, v, w} is not Linearly Independent in V.

Proof:

     Given V be a vector space.

     u, v, w are vectors in V.

    Let scalars a, b, c in R such that au+bv+cw=0

    au+bv+c(3u+2v)=0

    (a+3c)u+(b+2c)V=0

     a+3c=0 b+2c=0

     a = -3c, b=-2c 

   If c=0 then a=0, b=0.

  for \(c\neq 0\) the vectors {u, v, w} are not L.I.


Q2) Show that the system of Vectors (1,3,2), (1,-7,-8)(2,1,-1) of V is Linearly Dependent.

Solution:

    a(1,3,2)+b(1,-7,-8)+c(2,1,-1)=(0,0,0)

    a+b+2c=0 ------(1)

    3a-7b+c=0 -----(2)

    2a-8b-c =0 -----(3)

   solving (1), (2) and (3) we get

a=3, b=1, c=-2

Therefore given vectors are Linearly Dependent.


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