linearly independent equation [help]

[elec350]

hello
pls define the linearly independent equation and linearly dependent equation for me.

 

[mayyan]

the system is independent if the only sulotion to:

a1V1 + a2V2+...+AnVn = 0

a1,a2,An - constants
V1,V2,...,Vn - vecrtors representing the euations

is: a1=0,a2=0,..,An=0


example
2x1+3x2+0x3=0 => V1=(2,3,0)
4x1+6x2+0x3=0 => V2=(4,6,0)
4x1+5x2+1x3=0 => V3=(4,5,1)

here 2V1 -1V2+0V3=0 this means the system is dependent

1x1+3x2+7x3=0

 

[elec350]

hi mayyan
your definition has ambiguity. pls expln more.

 

[master_picengineer]

Hi elec350,
mayyan is right and I wanna add that linearly independent equation can be represented by matrix of a1, a2, a3...

 

[mayyan]

hope the attachment will help

 

[SWINI]

an equation which contain solutions independent and not equal to zero
like
a1b1+a2b2+.....=0
in this equation
a1b1 is independent from a2b2 ( this can be proved from wronskian theorem )
wronskian
w= det(a1b1,a2b2/d(a1b1)d(a2b2))
also
a1b1, a2b2 not equal to zero