190 views
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 ?
0
I think it should be infinitely many solutions.
0
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.
0
draw matrix A   and  A:B you will understand that if  Rank(A) < Rank(A:B)  is inconsistent

+1 vote

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)

selected

1
2