1.7: Supplementary Exercises for Chapter 1
- Page ID
- 58924
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We show in Chapter [chap:4] that the graph of an equation \(ax + by + cz = d\) is a plane in space when not all of \(a\), \(b\), and \(c\) are zero.
- By examining the possible positions of planes in space, show that three equations in three variables can have zero, one, or infinitely many solutions.
- Can two equations in three variables have a unique solution? Give reasons for your answer.
- No. If the corresponding planes are parallel and distinct, there is no solution. Otherwise they either coincide or have a whole common line of solutions, that is, at least one parameter.
Find all solutions to the following systems of linear equations.
- \( \begin{array}[t]{rlrlrlrcr} x_1 & + & x_2 & + & x_3 & - & x_4 & = & 3 \\ 3x_1 & + & 5x_2 & - & 2x_3 & + & x_4 & = & 1 \\ -3x_1 & - & 7x_2 & + & 7x_3 & - & 5x_4 & = & 7 \\ x_1 & + & 3x_2 & - & 4x_3 & + & 3x_4 & = & -5 \\ \end{array}\)
- \( \begin{array}[t]{rlrlrlrcr} x_1 & + & 4x_2 & - & x_3 & + & x_4 & = & 2 \\ 3x_1 & + & 2x_2 & + & x_3 & + & 2x_4 & = & 5 \\ x_1 & - & 6x_2 & + & 3x_3 & & & = & 1 \\ x_1 & + & 14x_2 & - & 5x_3 & + & 2x_4 & = & 3 \\ \end{array}\)
- \(x_1 = \frac{1}{10}(-6s - 6t + 16)\), \(x_2 = \frac{1}{10}(4s - t + 1)\), \(x_3 = s\), \(x_4 = t\)
In each case find (if possible) conditions on \(a\), \(b\), and \(c\) such that the system has zero, one, or infinitely many solutions.
\( \begin{array}[t]{rlrlrcr} x & + & 2y & - & 4z & = & 4 \\ 3x & - & y & + & 13z & = & 2 \\ 4x & + & y & + & a^2z & = & a + 3 \\ \end{array}\) \( \begin{array}[t]{rlrlrcr} x & + & y & + & 3z & = & a \\ ax & + & y & + & 5z & = & 4 \\ x & + & ay & + & 4z & = & a \end{array}\)
- If \(a = 1\), no solution. If \(a = 2\), \(x = 2 - 2t\), \(y = -t\), \(z = t\). If \(a \neq 1\) and \(a \neq 2\), the unique solution is \(x = \frac{8 - 5a}{3(a - 1)}\), \(y = \frac{-2 - a}{3(a - 1)}\), \(z = \frac{a + 2}{3}\)
Show that any two rows of a matrix can be interchanged by elementary row transformations of the other two types.
\(\left[ \begin{array}{c} R_1 \\ R_2 \end{array} \right] \to\)
\(\left[ \begin{array}{c} R_1 + R_2 \\ R_2 \end{array} \right] \to \left[ \begin{array}{c} R_1 + R_2 \\ -R_1 \end{array} \right] \to \left[ \begin{array}{c} R_2 \\ -R_1 \end{array} \right] \to \left[ \begin{array}{c} R_2 \\ R_1 \end{array} \right]\)
If \(ad \neq bc\), show that \(\left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]\) has reduced row-echelon form \(\left[ \begin{array}{rrr} 1 & 0 \\ 0 & 1 \end{array} \right]\).
Find \(a\), \(b\), and \(c\) so that the system
\[ \begin{array}{rlrlrcr} x & + & ay & + & cz & = & 0 \\ bx & + & cy & - & 3z & = & 1 \\ ax & + & 2y & + & bz & = & 5 \end{array} \nonumber \]
has the solution \(x = 3\), \(y = -1\), \(z = 2\).
\(a = 1\), \(b = 2\), \(c = -1\)
Solve the system
\[ \begin{array}{rlrlrcr} x & + & 2y & + & 2z & = & -3 \\ 2x & + & y & + & z & = & -4 \\ x & - & y & + & iz & = & i \end{array} \nonumber \]
where \(i^{2} = -1\). [See Appendix [chap:appacomplexnumbers].]
Show that the real system
\[\left \{ \begin{array}{rlrlrcr} x & + & y & + & z & = & 5 \\ 2x & - & y & - & z & = & 1 \\ -3x& + & 2y & + & 2z & = & 0 \end{array} \right. \nonumber \]
has a complex solution: \(x = 2\), \(y = i\), \(z = 3 - i\) where \(i^{2} = -1\). Explain. What happens when such a real system has a unique solution?
The (real) solution is \(x = 2\), \(y = 3 - t\), \(z = t\) where \(t\) is a parameter. The given complex solution occurs when \(t = 3 - i\) is complex. If the real system has a unique solution, that solution is real because the coefficients and constants are all real.
A man is ordered by his doctor to take \(5\) units of vitamin A, \(13\) units of vitamin B, and \(23\) units of vitamin C each day. Three brands of vitamin pills are available, and the number of units of each vitamin per pill are shown in the accompanying table.
Brand |
Vitamin |
||
A |
B |
C |
|
1 |
1 |
2 |
4 |
2 |
1 |
1 |
3 |
3 |
0 |
1 |
1 |
- Find all combinations of pills that provide exactly the required amount of vitamins (no partial pills allowed).
- If brands 1, 2, and 3 cost 3, 2, and 5 per pill, respectively, find the least expensive treatment.
- \(5\) of brand 1, \(0\) of brand 2, \(3\) of brand 3
A restaurant owner plans to use \(x\) tables seating \(4\), \(y\) tables seating \(6\), and \(z\) tables seating \(8\), for a total of \(20\) tables. When fully occupied, the tables seat \(108\) customers. If only half of the \(x\) tables, half of the \(y\) tables, and one-fourth of the \(z\) tables are used, each fully occupied, then \(46\) customers will be seated. Find \(x\), \(y\), and \(z\).
- Show that a matrix with two rows and two columns that is in reduced row-echelon form must have one of the following forms:
\[\left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right] \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right] \left[ \begin{array}{rr} 1 & * \\ 0 & 0 \end{array} \right] \nonumber \]
- List the seven reduced row-echelon forms for matrices with two rows and three columns.
- List the four reduced row-echelon forms for matrices with three rows and two columns.
An amusement park charges $\(7\) for adults, $\(2\) for youths, and $\(0.50\) for children. If \(150\) people enter and pay a total of $\(100\), find the numbers of adults, youths, and children. [Hint: These numbers are nonnegative integers.]
Solve the following system of equations for \(x\) and \(y\).
\[ \begin{array}{rlrlrcr} x^2 & + & xy & - & y^2 & = & 1 \\ 2x^2 & - & xy & + & 3y^2 & = & 13 \\ x^2 & + & 3xy & + & 2y^2 & = & 0 \\ \end{array} \nonumber \]
[Hint: These equations are linear in the new variables \(x_{1} = x^{2}\), \(x_{2} = xy\), and \(x_{3} = y^{2}\).]