Does a homogeneous system of 4 equations in 5 variables have infinitely many solutions????
Because the matrix corresponding to coefficients of the variables has its rank less than the number of variables. In such a case at least one variable will be free variable and hence can take infinitely many values from corresponding field.
Homogeneous systems of linear equations are systems where the constant term in each equations is zero. Therefore, they have the form [A|0], where A is the matrix of coefficients of the variables in the system of equations. Systems of this type always have a solution. There is always the trivial solution where [x1, x2, ..., xn] = [0,0,...0].
Apart from this ,we four equation in 5 variables, which means we have one free variables and we can assign any arbitrary value to it . This implies we have an infinite no. of solutions.