Page 32 - 35Linear Algebra

32 What is Linear Algebra?

says that the rate of change of f is proportional to f. It describes
exponential growth because the exponential function

f(t) = f(0)e 2t

satisfies the DE for any number f(0). The number 2 in the DE is called
the constant of proportionality. A similar DE

d 2
f = f
dt t

has a time-dependent “constant of proportionality”.

(a) Do you think that the second DE describes exponential growth?

(b) Write both DEs in the form Df = 0 with D a linear operator.

5. Pablo is a nutritionist who knows that oranges always have twice as
much sugar as apples. When considering the sugar intake of schoolchil-
dren eating a barrel of fruit, he represents the barrel like so:

fruit

(s, f)

sugar

Find a linear operator relating Pablo’s representation to the “everyday”
representation in terms of the number of apples and number of oranges.
Write your answer as a matrix.

Hint: Let λ represent the amount of sugar in each apple.

Hint

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