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while solving linear equations, there comes a case where rank < number of varibles,

then we say there are n-r linearly independent solution.

What exactly does it mean ?

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Rank<n (no of vars) 

Am considering homogeneous case,  

Same applies in nonhomogeneous when r(a) =r(a:b) <n

Linearly dependent equations eg x+y=2 and 2x+2y=4 

We have two equations but they r  linearly dependent 

So  actually they r equivalent to one equation. (here 

Also you ll hv to put x=k and get y value corresponding to it so 1 linearly independent soln ie x and y linearly dependent on x)  

In homogeneous, 

Infinite soln coz rank <no of vars

unique soln not possible (coz fr solving n vars we need n linearly independent equations) 

No of linearly  lndependent solutions =n-r

Rank= no of linearly independent equations 

coz n-r vars cannot be solved uniquely u hv to assign sm consts k, l.. Note k, l are not linearly dependent (no of such vars is n-r)

. For other vars  u ll get values in terms of l and k eg k+l, l/2  etc.(these are linearly dependent on k, l) 

 

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