91Ó°ÊÓ

A linear system of the form $$A \vec{x}=\overrightarrow{0}$$ is called homogeneous. Justify the following facts: a. All homogeneous systems are consistent. b. A homogeneous system with fewer equations than unknowns has infinitely many solutions. c. If \(\vec{x}_{1}\) and \(\vec{x}_{2}\) are solutions of the homogeneous sys\(\operatorname{tem} A \vec{x}=0,\) then \(\vec{x}_{1}+\vec{x}_{2}\) is a solution as well. d. If \(\vec{x}\) is a solution of the homogeneous system \(A \vec{x}=\overrightarrow{0}\) and \(k\) is an arbitrary constant, then \(k \vec{x}\) is a solution as well.

Consider the accompanying table. For some linear systems \(A \vec{x}=\vec{b},\) you are given either the rank of the coefficient matrix \(A,\) or the rank of the augmented matrix \([A | \vec{b}] .\) In each case, state whether the system could have no solution, one solution, or infinitely many solutions. There may be more than one possibility for some systems. Justify your answers. $$\begin{array}{lcccc} \hline & \begin{array}{c} \text { Number of } \\ \text { Equations } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Unknowns } \end{array} & \begin{array}{c} \text { Rank } \\ \text { of } \boldsymbol{A} \end{array} & \begin{array}{c} \text { Rank } \\ \text { of }[\boldsymbol{A}, \overrightarrow{\boldsymbol{b}}] \end{array} \\ \hline \text { a. } & 3 & 4 & \- & 2 \\ \text { b. } & 4 & 3 & 3 & \- \\ \text { c. } & 4 & 3 & \- & 4 \\ \text { d. } & 3 & 4 & 3 & \- \\ \hline \end{array}$$

\((0,0),(1,0),(2,0),(3,0),\) and (1,1)

Is the vector $$\left[\begin{array}{c} 30 \\ -1 \\ 38 \\ 56 \\ 62 \end{array}\right]$$ a linear combination of $$\left[\begin{array}{l} 1 \\ 7 \\ 1 \\ 9 \\ 4 \end{array}\right], \quad\left[\begin{array}{l} 5 \\ 6 \\ 3 \\ 2 \\ 8 \end{array}\right], \quad\left[\begin{array}{l} 9 \\ 2 \\ 3 \\ 5 \\ 2 \end{array}\right], \quad\left[\begin{array}{r} -2 \\ -5 \\ 4 \\ 7 \\ 9 \end{array}\right] ?$$

\((0,0),(1,0),(2,0),(0,1),(0,2),\) and (1,1)

For which values of the constant \(c\) is \(\left[\begin{array}{c}1 \\ c \\\ c^{2}\end{array}\right]\) a linear combination of \(\left[\begin{array}{l}1 \\\ 2 \\ 4\end{array}\right]\) and \(\left[\begin{array}{l}1 \\ 3 \\\ 9\end{array}\right] ?\)

"A rooster is worth five coins, a hen three coins, and 3 chicks one coin. With 100 coins we buy 100 of them. How many roosters, hens, and chicks can we buy?" (From the Mathematical Manual by Zhang Qiujian, Chapter \(3,\) Problem \(38 ;\) 5th century A.D.) Commentary: This famous Hundred Fowl Problem has reappeared in countless variations in Indian, Arabic, and European texts (see Exercises 73 through 76 ); it has remained popular to this day. See Exercise 46 of this section.

Make me a crown weighing 60 minae from a mixture of gold, bronze, tin, and wrought iron. Let the gold and bronze together form two-thirds of the weight, the gold and tin together three-fourths, and the gold and iron three-fifths. Tell me how much gold, tin, bronze, and iron you must use. (From the Greek Anthology by Metrodorus, \(6 \text { th century } A . D .)\)

3 cows graze 1 field bare in 2 days, 7 cows graze 4 fields bare in 4 days, and 3 cows graze 2 fields bare in 5 days. It is assumed that each field initially provides the same amount, \(x,\) of grass; that the daily growth, \(y,\) of the fields remains constant; and that all the cows eat the same amount, \(z,\) each day. (Quantities \(x, y,\) and \(z\) are measured by weight.) Find all the solutions of this problem. (This is a special case of a problem discussed by Isaac Newton in his Arithmetica Universalis, 1707 .)