TITLE: Solve a linear system with more variables than equations
QUESTION [14 upvotes]: Suppose that, after a series of elementary row operations the augmented matrix of a linear system with variables $x_1$, $x_2$, $x_3$, $x_4$ is transformed into reduced row echelon form as follows:
$$\left(\begin{array}{cccc|c}1 & 0 & 0 & 1 & 0\\0 & 1 & 0 & 2 & 1 \\0 & 0 & 1 & 3 & 0 \end{array}\right)$$.
Can I solve the linear system as below?
Let $t$ be an arbitrary real number. Then solving each linear equation corresponding to the augmented matrix for leading variable and setting $x_4=t$, we get $x_1=-t, x_2=1-2t$, and $x_3=-3t$. Thus the general solution of the linear system is
\begin{align}
x_1=-t\\
x_2=1-2t\\
x_3=-3t\\
\end{align}
where t is an arbitrary real number.

REPLY [5 votes]: What you have done is a decomposition. Given an underconstrained system $A x =b$ you can split the variables to dependent and independent components such that
$$ x = T x_{dep} + U x_{ind} $$
in your example
$$ \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} =
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}
\begin{pmatrix} x_1 \\ x_2 \\ x_3  \end{pmatrix} +
\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \begin{pmatrix} x_4  \end{pmatrix} $$
Now you reduce the system with
$$ \begin{aligned} 
A T x_{dep} + A U x_{ind} & = b \\
A T x_{dep}  & = b - A U x_{ind} \\
x_{dep} &= (A T)^{-1} \left(b - A U x_{ind}\right)
\end{aligned} $$
The final values are
$$ x = T (A T)^{-1} b + \left({\bf 1_{4×4}} - T (A T)^{-1} A\right) U x_{dep} \\
   x = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1  \end{bmatrix} \begin{pmatrix}0\\1\\0\end{pmatrix} + \begin{bmatrix} 0&0&0&-1 \\0&0&0&-2 \\ 0&0&0&-3 \\ 0&0&0&1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \begin{pmatrix} x_4  \end{pmatrix} $$
$$
 x = \begin{pmatrix} -x_4 \\ 1-2 x_4 \\ -3 x_4 \\ x_4 \end{pmatrix}
$$
{"set_name": "stack_exchange", "score": 14, "question_id": 602927}