Linear Algebra

20 Chapter One. Linear Systems(b) Answer the above question for the system.ax + y = a 3x + ay = 1[USSR Olympiad no. 174]? 2.31 In air a gold-surfaced sphere weighs 7588 grams. It is known that it maycontain one or more of the metals aluminum, copper, silver, or lead. When weighedsuccessively under standard conditions in water, benzene, alcohol, and glycerineits respective weights are 6588, 6688, 6778, and 6328 grams. How much, if any,of the forenamed metals does it contain if the specific gravities of the designatedsubstances are taken to be as follows?Aluminum 2.7 Alcohol 0.81Copper 8.9 Benzene 0.90Gold 19.3 Glycerine 1.26Lead 11.3 Water 1.00Silver 10.8[Math. Mag., Sept. 1952]I.3 General = Particular + HomogeneousThe prior subsection has many descriptions of solution sets. They all fit apattern. They have a vector that is a particular solution of the system addedto an unrestricted combination of some other vectors. The solution set fromExample 2.13 illustrates.⎛ ⎞ ⎛ ⎞ ⎛ ⎞0 1 1/24⎟ −1{⎜0⎟ + w⎜ 3⎟⎝0⎠⎝ 1 ⎠ + u −1⎜1/2∣⎟ w, u ∈ R}⎝ 0 ⎠0} {{ } }0{{1}particularsolutionunrestrictedcombinationThe 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 can be used to form combinations.That example shows an infinite solution set conforming to the pattern. Wecan think of the other two kinds of solution sets as also fitting the same pattern.A one-element solution set fits in that it has a particular solution, andthe unrestricted combination part is a trivial sum (that is, instead of being acombination of two vectors, as above, or a combination of one vector, it is acombination of no vectors). A zero-element solution set fits the pattern sincethere is no particular solution, and so the set of sums of that form is empty.We will show that the examples from the prior subsection are representative,in that the description pattern discussed above holds for every solution set.