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Adding apples to oranges

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Construction.

Any function to ${\bf R}$ (or any vector space) can be turned into a linear operator, if you can expand the domain.

How?

Question: We don't compare apples to oranges...

But can we add them?

Does "apple $+$ orange" make sense?

Answer: NO.

(Yes, if both are fruit then you get two fruit. blah blah, doesn't matter...)

In fact, this should be a very emphatic NO -- in calculus and before (or when speaking to non-mathematicians).

But let's try anyway:

$3$ apples + $5$ oranges = $3$ apples + $5$ oranges.

Hmm...

This is a linear combination of apples and oranges.

Consider this:

$(3a + 5o) + (11a - 7o) = 14a - 2.o$

Discovery: We can do algebra with these!

So, mathematically, I started with apples and oranges, and then I've built a vector space with ${\rm basis} = {\rm apple, orange}$:

AppleOrangeVectorSpace.png

$V = {\rm span}\{o,a\}$

More generally, given any set $S$, I can create a vector space $V$ with basis $S$, as the set of "formal" (finite) linear combinations of elements of $S$.

What about a function on $S$? Can we make it linear now?

Yes, if $$F \colon S \rightarrow {\bf R}$$ is any function, let's consider $$F^* \colon span(S) \rightarrow {\bf R}$$ given by $$F^*\left( \displaystyle\sum_{\stackrel{i=1}{s_i \in S}}^n \alpha_i s_i \right) = \displaystyle\sum_{\stackrel{i=1}{s_i \in S}}^n \alpha_i F(s_i)$$

The rule of Linear algebra is: Every function on a basis gives a linear operator on the whole space.