Page 417 - 35Linear Algebra

G.12 Diagonalization 417

fruit

(s, f)

sugar

WARNING: To make sense of what comes next you need to allow for the possibity
of a negative amount of fruit or sugar. This would be just like a bank, where
if money is owed to somebody else, we can use a minus sign.
The vector ~x says what is in the barrel and does not depend which mathe-
matical description is employed. The way nutritionists label ~x is in terms of
~
~
a pair of basis vectors f 1 and f 2 :

s
~ ~ ~ ~
~x = sf 1 + ff 2 = f 1 f 2 .
f
Thus our vector space now has a bunch of interesting vectors:

The vector ~x labels generally the contents of the barrel. The vector ~e 1 corre-
sponds to one apple and one orange. The vector ~e 2 is one orange and no apples.
~
The vector f 1 means one unit of sugar and zero total fruit (to achieve this
~
you could lend out some apples and keep a few oranges). Finally the vector f 2
represents a total of one piece of fruit and no sugar.
You might remember that the amount of sugar in an apple is called λ while
oranges have twice as much sugar as apples. Thus

s = λ (x + 2y)
f = x + y .

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