2017-08-12 07:46:30
Suppose that X1=-1, x2=2, x3=4, x4=-3 is a solution of a non-homogeneous linear system Ax = b and that the solution set of the homogeneous system Ax=0 is given by the formulas:x1= -3r +4s, x2=r-s, x3=r, x3=sa) Find the vector form of the general solution of Ax=bb) Find the vector form of the general solution Ax=0
2017-08-12 09:21:08

The answers will be as simple as writing the component solutions as components of a vector. For instance, [latex]mathbf{Ax}=mathbf 0[/latex] is solved by any vector belonging to the set [latex]left{mathbf xinmathbb R^4~:~x_1=-3r+4s,x_2=r-s,x_3=r,x_4=s,(r,s)inmathbb R^2 ight}[/latex] or simply as the vector [latex]mathbf x=egin{bmatrix}-3r+4s\r-s\r\send{bmatrix}[/latex] The general solution to [latex]mathbf{Ax}=mathbf b[/latex] will be the particular solution plus any vector belonging to the nullspace, so that the general solution would take the form [latex]mathbf x=egin{bmatrix}-3r+4s\r-s\r\send{bmatrix}+egin{bmatrix}-1\2\4\-3end{bmatrix}[/latex] [latex]mathbf x=egin{bmatrix}-1-3r+4s\2+r-s\4+r\-3+send{bmatrix}[/latex] where [latex]r,s[/latex] are any real numbers.