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)