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Linearly Dependent

MATHMAA

# 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

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

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.