Find the echelon form U, the free variables, and the special...

Find the echelon form $U$, the free variables, and the special solutions: $$ A=\left[\begin{array}{llll} 0 & 1 & 0 & 3 \\ 0 & 2 & 0 & 6 \end{array}\right], \quad b=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ $A x=$ ______ b is consistent (has a solution) when $b$ satisfies $b_{2}=$ Find the complete solution in the same form as equation (4).

Find the echelon form $U$, the free variables, and the special solutions:
$$
A=\left[\begin{array}{llll}
0 & 1 & 0 & 3 \\
0 & 2 & 0 & 6
\end{array}\right], \quad b=\left[\begin{array}{l}
b_{1} \\
b_{2}
\end{array}\right]
$$
$A x=$ ______ b is consistent (has a solution) when $b$ satisfies $b_{2}=$ Find the complete solution in the same form as equation (4).

Key Concepts

Consistency of Linear Systems

The consistency of a system of linear equations indicates whether at least one solution exists. A system is consistent if there is no contradiction (for example, a row that simplifies to an impossibility like 0 = nonzero). In augmented matrices, the consistency condition can impose specific constraints on the entries of the vector on the right-hand side.

Special Solutions

Special solutions, sometimes referred to as particular solutions, are specific solutions obtained by assigning particular values to the free variables in an underdetermined system. These solutions, together with the general solution of the corresponding homogeneous system, allow the complete description of all possible solutions in a parametric vector form.

Row Echelon Form

The row echelon form of a matrix is achieved using elementary row operations so that all nonzero rows are above any zero rows and each leading coefficient (the first nonzero number from the left in a nonzero row) appears to the right of the leading coefficient of the row above it. This form simplifies solving linear systems by clearly identifying pivot positions and facilitating back-substitution.

Free Variables

Free variables occur in a linear system when there are columns in the row echelon form that do not contain a pivot. These variables are not directly solved by the system and can take arbitrary values. Their presence often indicates that a system has infinitely many solutions, and they are used to parametrize these solutions.

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