Consider the tall system \(\left(\begin{array}{rr|r}-1\amp 1\amp 1\\-2\amp 1\amp 1\\-3\amp 1\amp 2 \end{array} \right).\)
1.
Write the explicit representation of the system, using \(\left(\begin{array}{c}x\\y\end{array}\right).\)
2.
Write the slope-intercept form of each of the equations in the explicit representation.
3.
Use elimination to show that the system is inconsistent.
4.
Graph the three lines carefully and interpret the inconsistency of the system in terms of your graph.
Exercise Group.
Fill in the table with the number of solutions (none/\(0,\) unique/\(1,\) or infinite/\(\infty\)) for each case of the system \(\A\x=\b\text{:}\)
5.
For general \(\A\x=\b:\)
\begin{equation*}
\begin{array}{l|c|c}\amp\text{ is consistent}\amp\text{ is inconsistent}\\\hline\text{ is singular}\amp\amp\\\hline\text{ is nonsingular}\amp\amp\end{array}
\end{equation*}
6.
For consistent \(\A\x=\b:\)
\begin{equation*}
\begin{array}{l|c|c}\amp \,\,m\lt n\,\,\amp \,\,m=n\,\,\\\hline \U\x=\c\text{ has a full set of pivots}\amp\amp\\\hline\U\x=\c\text{ does not have a full set of pivots}\amp\amp\end{array}
\end{equation*}
Exercise Group.
If possible, reduce the system to an upper-triangular system by elimination then solve the system for \(\vec{x}=\begin{pmatrix}x_1\\x_2\\x_3 \end{pmatrix} \text{.}\) If this is not possible, explain why. Be sure to label your row operations vertically adjacent to your arrows.
Find \(a,b,c,d\in\R\setminus\{0\}\) with \(a\ne1,5\) so that \(\left(\begin{array}{rrr|r}1\amp 2\amp 3\amp 4\\5\amp 6\amp 7\amp 8\\a\amp b\amp c\amp d \end{array} \right)\) is singular and consistent. If this is not possible, explain why.
12.
Find \(a,b,c,d\in\R\setminus\{0\}\) with \(a\ne1,5\) so that \(\left(\begin{array}{rrr|r}1\amp 2\amp 3\amp 4\\5\amp 6\amp 7\amp 8\\a\amp b\amp c\amp d \end{array} \right)\) is singular and inconsistent. If this is not possible, explain why.
Exercise Group.
If there are two distinct solutions to \(A\vec{x}=\vec{b},\) there must be three distinct solutions.
To see this, let \(\vec{x_1}\) and \(\vec{x_2}\) be distinct solutions to \(A\vec{x}=\vec{b}.\) We create a third solution \(\vec{x_3}\) distinct from both \(\vec{x_1}\) and \(\vec{x_2}\) as follows.
which means that \(\vec{x_1},\vec{x_2},\) and \(4\vec{x_1}-3\vec{x_2}\) are distinct, at least as long as \(4\vec{x_1}-3\vec{x_2}\) does not equal either of \(\vec{x_1},\vec{x_2}.\) Below you will 1) prove this fact and 2) generalize the above coefficients \(4,-3\text{,}\) leading to an infinite set
of distinct solutions parametrized by the real variable \(r.\)
13.
Prove by contradiction and cases that \(4\vec{x_1}-3\vec{x_2}\) does not equal either of \(\vec{x_1},\vec{x_2}\) in the third case of Theorem 3.1.4. Be sure to assume that \(\vec{x_1}\ne\vec{x_2}.\)
14.
Let’s explore the third case of Theorem 3.1.4 a little more deeply. Suppose \(\vec{y}\) and \(\vec{z}\) both solve the equation \(\vec{x}=\vec{b}\text{.}\) Let \(s,t\in\R\) and compute \(A\left(s\vec{y}+t\vec{z}\right)\text{.}\)
15.
Find a condition that \(s,t\) must satisfy for \(A\left(s\vec{y}+t\vec{z}\right)=\vec{b}\) to hold. Why does this condition guarantee that there are an infinite number of solutions to \(A\vec{x}=\vec{b}\text{?}\)
