Linear Algebra

20 Chapter One. Linear Systems(b) a two-parameter solution set;(c) a three-parameter solution set.? 2.31 [Shepelev] This puzzle is from a Russian web-site http://www.arbuz.uz/, andthere are many solutions to it, but mine uses linear algebra and is very naive.There’s a planet inhabited by arbuzoids (watermeloners, if to translate fromRussian). Those creatures are found in three colors: red, green and blue. Thereare 13 red arbuzoids, 15 blue ones, and 17 green. When two differently coloredarbuzoids meet, they both change to the third color.The question is, can it ever happen that all of them assume the same color?? 2.32 [USSR Olympiad no. 174](a) Solve the system of equations.ax + y = a 2x + ay = 1For what values of a does the system fail to have solutions, and for what valuesof a are there infinitely many solutions?(b) Answer the above question for the system.ax + y = a 3x + ay = 1? 2.33 [Math. Mag., Sept. 1952] In air a gold-surfaced sphere weighs 7588 grams. Itis known that it may contain one or more of the metals aluminum, copper, silver,or lead. When weighed successively under standard conditions in water, benzene,alcohol, and glycerin its respective weights are 6588, 6688, 6778, and 6328 grams.How much, if any, of the forenamed metals does it contain if the specific gravitiesof the designated substances are taken to be as follows?Aluminum 2.7 Alcohol 0.81Copper 8.9 Benzene 0.90Gold 19.3 Glycerin 1.26Lead 11.3 Water 1.00Silver 10.8I.3 General = Particular + HomogeneousIn the prior subsection the descriptions of solution sets all fit a pattern. Theyhave a vector that is a particular solution of the system added to an unrestrictedcombination of some other vectors. The solution set from Example 2.13illustrates. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞01 1/2{40⎜ ⎟⎝0⎠0particularsolution−1−1+ w3+ u1/2⎜ ⎟ ⎜ ⎟⎝ 1⎠⎝ 0⎠0 1unrestrictedcombination∣ w, u ∈ R}The combination is unrestricted in that w and u can be any real numbers —there is no condition like “such that 2w − u = 0” that would restrict which pairsw, u we can use.