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5DUQ3-Y_gX4 | That is the question. |
5DUQ3-Y_gX4 | The graph is a line. |
5DUQ3-Y_gX4 | So we all think of it
as linear, but go ahead. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Yeah. |
5DUQ3-Y_gX4 | So does it satisfy the rules? |
5DUQ3-Y_gX4 | That's the question. |
5DUQ3-Y_gX4 | So if I multiply the
input by 2, does it |
5DUQ3-Y_gX4 | multiply the output by 2? |
5DUQ3-Y_gX4 | No. |
5DUQ3-Y_gX4 | So if we do O of 2-- |
5DUQ3-Y_gX4 | no, let's do 3. |
5DUQ3-Y_gX4 | 3v, that's, what, 6v plus 1. |
5DUQ3-Y_gX4 | And that's very much
not equal to 3Ov, |
5DUQ3-Y_gX4 | which that would be 6v plus 3. |
5DUQ3-Y_gX4 | So this one, it
does have a name. |
5DUQ3-Y_gX4 | It's related. |
5DUQ3-Y_gX4 | These are sometimes
called affine. |
5DUQ3-Y_gX4 | ALAN EDELMAN: Affine, but
not linear even if the graph |
5DUQ3-Y_gX4 | is demonstrably aligned? |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Yeah. |
5DUQ3-Y_gX4 | ALAN EDELMAN: They're not linear
in the sense of linear algebra. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Right? |
5DUQ3-Y_gX4 | So another one is clearly
multiplication by a matrix. |
5DUQ3-Y_gX4 | That's why we do matrices
in linear algebra. |
5DUQ3-Y_gX4 | Because they're a nice way of
writing down a linear operation |
5DUQ3-Y_gX4 | if your vs are column vectors. |
5DUQ3-Y_gX4 | They're not the only way of
writing down linear operation. |
5DUQ3-Y_gX4 | So for example, if you
take a column vector |
5DUQ3-Y_gX4 | and multiply it by
3, you could write |
5DUQ3-Y_gX4 | that down as a matrix with
all 3s along the diagonal. |
5DUQ3-Y_gX4 | But it's a lot easier
to write that down as 3, |
5DUQ3-Y_gX4 | I'd say, as a scalar, than
to write it down as a matrix. |
5DUQ3-Y_gX4 | So another example,
just to be more-- |
5DUQ3-Y_gX4 | so another vector space. |
5DUQ3-Y_gX4 | If you took 1806, you learned
that we can have a-- whoops, |
5DUQ3-Y_gX4 | I have to get my color scheme. |
5DUQ3-Y_gX4 | Yeah, so my vectors are red. |
5DUQ3-Y_gX4 | Suppose the vector space V is
the set of functions f of x |
5DUQ3-Y_gX4 | that take real numbers in and
give you real numbers out. |
5DUQ3-Y_gX4 | Those are a perfectly
good vector space. |
5DUQ3-Y_gX4 | I can take two functions. |
5DUQ3-Y_gX4 | I can add them or subtract
them, get another function. |
5DUQ3-Y_gX4 | I can take a function
and multiply by 2, |
5DUQ3-Y_gX4 | get another function. |
5DUQ3-Y_gX4 | ALAN EDELMAN: Wait, how do
we get sine plus cosine? |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: I get
sine x plus cosine x. |
5DUQ3-Y_gX4 | It's just got some
other function. |
5DUQ3-Y_gX4 | So if you take sine
x plus cosine x, |
5DUQ3-Y_gX4 | that's the function f of x
equals sine x plus cosine x. |
5DUQ3-Y_gX4 | It's another rule
that gives you-- takes |
5DUQ3-Y_gX4 | real numbers to real numbers. |
5DUQ3-Y_gX4 | And so what would be your
linear operators on this? |
5DUQ3-Y_gX4 | Well, multiplication by a
scalar, that, of course, works. |
5DUQ3-Y_gX4 | So let's think of L on a
function f of x is just 2 f |
5DUQ3-Y_gX4 | of x. |
5DUQ3-Y_gX4 | That takes a function
in, function out. |
5DUQ3-Y_gX4 | That's linear. |
5DUQ3-Y_gX4 | What about a linear operator
on a function of f of x? |
5DUQ3-Y_gX4 | Again, that gives you
the derivative, just |
5DUQ3-Y_gX4 | the ordinary 18.01 derivative. |
5DUQ3-Y_gX4 | This is the 18.01 derivative. |
5DUQ3-Y_gX4 | Obviously, that only works if
the function is differentiable. |
5DUQ3-Y_gX4 | So maybe we can
look at the subspace |
5DUQ3-Y_gX4 | of differentiable functions. |
5DUQ3-Y_gX4 | That's also a vector space. |
5DUQ3-Y_gX4 | Because if I take two
differentiable functions |
5DUQ3-Y_gX4 | and add or subtract or
multiply by constants, |
5DUQ3-Y_gX4 | they're still differentiable. |
5DUQ3-Y_gX4 | I could also do integration. |
5DUQ3-Y_gX4 | So if f of x that takes
a function f of x in |
5DUQ3-Y_gX4 | and gives you the integral
from, I don't know, |
5DUQ3-Y_gX4 | 0 to x of f prime-- |
5DUQ3-Y_gX4 | no, so f of x prime and dx
prime if they're integrable. |
5DUQ3-Y_gX4 | Again, we need to
restrict what functions |
5DUQ3-Y_gX4 | are allowed if we're taking
derivatives or integrals, so |
5DUQ3-Y_gX4 | things where these exist. |
5DUQ3-Y_gX4 | But this is perfectly linear. |
5DUQ3-Y_gX4 | Why? |
5DUQ3-Y_gX4 | Because if I take the
function and I double it, |
5DUQ3-Y_gX4 | if I double the integrand,
it doubles the integrals. |
5DUQ3-Y_gX4 | If I add two integrands,
you add the integrals. |
5DUQ3-Y_gX4 | Integration is a
linear operation. |
5DUQ3-Y_gX4 | Derivative is a
linear operation. |
5DUQ3-Y_gX4 | Another fun one is suppose
we take L of f of x. |
5DUQ3-Y_gX4 | And the output is the
function f of x squared. |
5DUQ3-Y_gX4 | So this doesn't look linear. |
5DUQ3-Y_gX4 | I have a square there. |
5DUQ3-Y_gX4 | But why is this linear? |
5DUQ3-Y_gX4 | Why? |
5DUQ3-Y_gX4 | Because, let's see, if I
take L of two functions, |
5DUQ3-Y_gX4 | if I have f of x plus g of x,
that should be f of x squared |
5DUQ3-Y_gX4 | plus g of x squared. |
5DUQ3-Y_gX4 | I'm squaring the
input, not the output. |
5DUQ3-Y_gX4 | So that's equal to
L of f plus L of g. |
5DUQ3-Y_gX4 | ALAN EDELMAN: So
I'll just comment. |
5DUQ3-Y_gX4 | Leave it to mathematicians to
take what most people would |
5DUQ3-Y_gX4 | think of as just a
column of numbers |
5DUQ3-Y_gX4 | and abstract it out and say that
this finite dimensional column |
5DUQ3-Y_gX4 | of numbers is somehow the
same as continuous functions |
5DUQ3-Y_gX4 | or differentiable functions,
satisfies the same axioms. |
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