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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 ?
asked in Linear Algebra by Loyal (7.1k points) | 196 views
I think it should be infinitely many solutions.
that is not my doubt.

I know once r < n then

In homogeneous equation there will be non trivial infinite many solution  and

In non homogeneous, it  is possible to have two cases

      (i) r(A) = r(A|B) <n, infinite many solutions possible &

     (ii) r(A) <r(A|B) & r(A) <n them AX = B is inconsistent i.e. no solution.

But whenever there is a infinite solution possible we say there are (n-r) linearly independent solution.
draw matrix A   and  A:B you will understand that if  Rank(A) < Rank(A:B)  is inconsistent

1 Answer

+1 vote
Best answer

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) 


answered by Loyal (8.1k points)
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