Question

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=s

a) Find the vector form of the general solution of Ax=b

b) Find the vector form of the general solution Ax=0

Answer 1

The answers will be as simple as writing the component solutions as components of a vector. For instance, $\mathbf{Ax}=\mathbf 0$ is solved by any vector belonging to the set

$\left\{\mathbf x\in\mathbb R^4~:~x_1=-3r+4s,x_2=r-s,x_3=r,x_4=s,(r,s)\in\mathbb R^2\right\}$

or simply as the vector

$\mathbf x=\begin{bmatrix}-3r+4s\\r-s\\r\\s\end{bmatrix}$

The general solution to $\mathbf{Ax}=\mathbf b$ will be the particular solution plus any vector belonging to the nullspace, so that the general solution would take the form

$\mathbf x=\begin{bmatrix}-3r+4s\\r-s\\r\\s\end{bmatrix}+\begin{bmatrix}-1\\2\\4\\-3\end{bmatrix}$
$\mathbf x=\begin{bmatrix}-1-3r+4s\\2+r-s\\4+r\\-3+s\end{bmatrix}$

where $r,s$ are any real numbers.