In the spirit of Example ISSI, describe the infinite solution set for Archetype J.
For Exercises C21–C28, find the solution set of the system of linear equations. Give the values of $n$ and $r$, and interpret your answers in light of the theorems of this section.
\begin{align*} x_1 + 4x_2 + 3x_3 - x_4 &= 5\\ x_1 - x_2 + x_3 + 2x_4 &= 6\\ 4x_1 + x_2 + 6x_3 + 5x_4 &= 9 \end{align*}
\begin{align*} x_1 - 2x_2 + x_3 - x_4 &= 3\\ 2x_1 - 4x_2 + x_3 + x_4 &= 2\\ x_1 - 2x_2 - 2x_3 + 3x_4 &= 1 \end{align*}
\begin{align*} x_1 - 2x_2 + x_3 - x_4 &= 3\\ x_1 + x_2 + x_3 - x_4 &= 1\\ x_1 \quad + x_3 - x_4 &= 2 \end{align*}
\begin{align*} x_1 - 2x_2 + x_3 - x_4 &= 2\\ x_1 + x_2 + x_3 - x_4 &= 2\\ x_1 \quad + x_3 - x_4 &= 2 \end{align*}
\begin{align*} x_1 + 2x_2 + 3x_3 &= 1\\ 2x_1 - x_2 + x_3 &= 2\\ 3x_1 + x_2 + x_3 &= 4\\ \quad x_2 + 2x_3 &= 6 \end{align*}
\begin{align*} x_1 + 2x_2 + 3x_3 &= 1\\ 2x_1 - x_2 + x_3 &= 2\\ 3x_1 + x_2 + x_3 &= 4\\ \quad 5x_2 + 2x_3 &= 1 \end{align*}
\begin{align*} x_1 + 2x_2 + 3x_3 &= 0\\ 2x_1 - x_2 + x_3 &= 2\\ x_1 - 8x_2 - 7x_3 &= 1\\ \quad x_2 + x_3 &= 0 \end{align*}
\begin{align*} x_1 + 2x_2 + 3x_3 &= 1\\ 2x_1 - x_2 + x_3 &= 2\\ x_1 - 8x_2 - 7x_3 &= 1\\ \quad x_2 + x_3 &= 0 \end{align*}
The details for Archetype J include several sample solutions. Verify that one of these solutions is correct (any one, but just one). Based only on this evidence, and especially without doing any row operations, explain how you know this system of linear equations has infinitely many solutions.
Consider Archetype J, and specifically the row-reduced version of the augmented matrix of the system of equations, denoted as $B$ here, and the values of $r$, $D$ and $F$ immediately following. Determine the values of the entries \begin{align*} \matrixentry{B}{1,d_1}& & \matrixentry{B}{3,d_3}& & \matrixentry{B}{1,d_3}& & \matrixentry{B}{3,d_1}& & \matrixentry{B}{d_1,1}& & \matrixentry{B}{d_3,3}& & \matrixentry{B}{d_1,3}& & \matrixentry{B}{d_3,1}& & \matrixentry{B}{1,f_1}& & \matrixentry{B}{3,f_1}& \end{align*} (See Exercise TSS.M70 for a generalization.)
For Exercises M51–M57 say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions.
A system with 8 equations and 6 variables. The reduced row-echelon form of the augmented matrix of the system has 7 pivot columns.
Without doing any computations, and without examining any solutions, say as much as possible about the form of the solution set for each archetype that is a system of equations.
Archetype A,
Archetype B,
Archetype C,
Archetype D,
Archetype E,
Archetype F,
Archetype G,
Archetype H,
Archetype I,
Archetype J
Suppose that $B$ is a matrix in reduced row-echelon form that is equivalent to the augmented matrix of a system of equations with $m$ equations in $n$ variables. Let $r$, $D$ and $F$ be as defined in Definition RREF. What can you conclude, in general, about the following entries? \begin{align*} \matrixentry{B}{1,d_1}& & \matrixentry{B}{3,d_3}& & \matrixentry{B}{1,d_3}& & \matrixentry{B}{3,d_1}& & \matrixentry{B}{d_1,1}& & \matrixentry{B}{d_3,3}& & \matrixentry{B}{d_1,3}& & \matrixentry{B}{d_3,1}& & \matrixentry{B}{1,f_1}& & \matrixentry{B}{3,f_1}& \end{align*} If you cannot conclude anything about an entry, then say so. (See Exercise TSS.M46.)
An inconsistent system may have $r>n$. If we try (incorrectly!) to apply Theorem FVCS to such a system, how many free variables would we discover?
Suppose $A$ is the augmented matrix of a system of linear equations in $n$ variables. and that $B$ is a row-equivalent matrix in reduced row-echelon form with $r$ pivot columns. If $r=n+1$, prove that the system of equations is inconsistent.
Suppose that $B$ is a matrix in reduced row-echelon form that is equivalent to the augmented matrix of a system of equations with $m$ equations in $n$ variables. Let $r$, $D$ and $F$ be as defined in Definition RREF. Prove that $d_k\geq k$ for all $1\leq k\leq r$. Then suppose that $r\geq 2$ and $1\leq k <\ell\leq r$ and determine what can you conclude, in general, about the following entries. \begin{align*} \matrixentry{B}{k,d_k}& & \matrixentry{B}{k,d_\ell}& & \matrixentry{B}{\ell,d_k}& & \matrixentry{B}{d_k,k}& & \matrixentry{B}{d_k,\ell}& & \matrixentry{B}{d_\ell,k}& & \matrixentry{B}{d_k,f_\ell}& & \matrixentry{B}{d_\ell,f_k}& \end{align*} If you cannot conclude anything about an entry, then say so. (See Exercise TSS.M46 and Exercise TSS.M70.)
Suppose that the coefficient matrix of a consistent system of linear equations has two columns that are identical. Prove that the system has infinitely many solutions.
Consider the system of linear equations $\linearsystem{A}{\vect{b}}$, and suppose that every element of the vector of constants $\vect{b}$ is a common multiple of the corresponding element of a certain column of $A$. More precisely, there is a complex number $\alpha$, and a column index $j$, such that $\vectorentry{\vect{b}}{i}=\alpha\matrixentry{A}{ij}$ for all $i$. Prove that the system is consistent.