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Print
Worksheet 3.5.1 Worksheet
Exercise Group.
Find the solution set to \(\A\x=\b\) for each of the following, citing any theorems you use.
For any system having exactly one solution, write the linear combination of the columns of \(\A\) which equals \(\b.\)
For any system having more than one solution, choose different values of the parameters in the element form of the solution set to find three distinct linear combinations of the columns of \(\A\) which are equal to \(\b.\)
1. \(\A=\begin{pmatrix} 1\amp 2\amp 3\amp 4\\5\amp 6\amp 7\amp 8\end{pmatrix},\quad\b=\left(\begin{array}{r}1\\-3 \end{array} \right)\) 2. \(\A=\left(\begin{array}{rrr}1\amp -2\amp 4\\2\amp -3\amp 5\\3\amp -4\amp 6 \end{array} \right),\quad\b=\left(\begin{array}{r}2\\3\\7 \end{array} \right)\) 3. \(\A=\left(\begin{array}{rrrr}1\amp 0\amp 5\amp -7\\1\amp 1\amp 0\amp 3\\2\amp 1\amp 5\amp -4 \end{array}\right),\quad\b=\left(\begin{array}{r}1\\2\\3 \end{array}\right)\) 4. \(\A=\left(\begin{array}{rrr}1\amp 2\amp -1\\1\amp 3\amp 1\\3\amp 8\amp 5 \end{array} \right),\quad\b=\left(\begin{array}{r}3\\5\\17 \end{array} \right)\) 5. \(\A=\left(\begin{array}{rrr}1\amp -2\amp 4\\2\amp -3\amp 5\\3\amp -5\amp 8\\1\amp 0\amp -1 \end{array} \right),\quad\b=\left(\begin{array}{r}2\\3\\7\\5 \end{array} \right)\) 6. \(\A=\left(\begin{array}{rrrr}1\amp 4\amp -2\amp -1\\-3\amp -11\amp 12\amp 2\\-1\amp -3\amp 8\amp 0 \end{array} \right),\quad\b=\left(\begin{array}{r}3\\-11\\-5 \end{array} \right)\) 7. \(\A=\left(\begin{array}{rrrrr}1\amp -2\amp 4\amp 2\amp 0\\3\amp -1\amp 5\amp 6\amp -3\\5\amp -5\amp 14\amp 14\amp -6\\-1\amp -3\amp 3\amp -2\amp 3 \end{array} \right),\quad\b=\left(\begin{array}{r}3\\-1\\1\\7 \end{array} \right)\)
8. Consider the matrix \(\A=\left(\begin{array}{rrrr}1\amp -4\amp 1\amp 3\\2\amp 3\amp -2\amp -1\\-1\amp 2\amp -3\amp -5 \end{array} \right).\)
For this \(\A\) and \(S\subset\R^4\) we may interpret the operation \(\x\mapsto\A\x,\,\x\in S\) as a function whose range is \(\left\{\A\x\,|\x\in S\right\}\text{.}\)
For each set below, multiply the general element of the set by \(\A,\) simplify, and express the range \(\left\{\A\x\,|\x\in S\right\}\) of the function \(\x\mapsto \A\x\) as concisely as possible, using element form if necessary.
\(\displaystyle S_1=\left\{\left.\left(\begin{array}{r}-3\\2\\4\\1\end{array}\right)+a\left(\begin{array}{r}1\\3\\8\\-7\end{array}\right)\,\right|\,a\in\R\right\}\)
\(\displaystyle S_2=\left\{\left.\left(\begin{array}{r}2\\1\\3\\-2\end{array}\right)+a\left(\begin{array}{r}-5\\1\\-6\\5\end{array}\right)\,\right|\,a\in\R\right\}\)
Explain the dramatic difference between the ranges \(\left\{\A\x\,|\x\in S_1\right\}\) and \(\left\{\A\x\,|\x\in S_2\right\}.\)
