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MATHMAA

# Linearly Independent

Linear Independence of Vectors :

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 Linearly Independent (L.I) if every relation of the form

$$a_{1}\alpha _{1}+a_{2}\alpha _{2}+\cdots +a_{n}\alpha _{n}=\bar{0},a_{i}'s \epsilon F$$

$$\Rightarrow a_{1}= 0, a_{2}= 0,\cdots a_{n}= 0$$

1) Show that the system of Vectors {(1,2,0), (0,3,1), (-1,0,1)} is Linearly Independent.

Solution:

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

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

a-c=0 , 2a+3b=0, b+c=0

a=c   , 2(c)+3b=0 , b+c =0

a=0, b=0, c=0

Therefore given vectors are Linearly Independent.