video_id stringclasses 7
values | text stringlengths 2 29.3k |
|---|---|
5DUQ3-Y_gX4 | I'm not asking for df. |
5DUQ3-Y_gX4 | Of course, they're related. |
5DUQ3-Y_gX4 | And let's see if we can
move this out of the way. |
5DUQ3-Y_gX4 | Good. |
5DUQ3-Y_gX4 | So what was I saying? |
5DUQ3-Y_gX4 | Yes. |
5DUQ3-Y_gX4 | So now, what I want
to do is basically use |
5DUQ3-Y_gX4 | this as the definition of a
derivative, a more general |
5DUQ3-Y_gX4 | definition of a derivative. |
5DUQ3-Y_gX4 | So the linear algebra
notion is that we have |
5DUQ3-Y_gX4 | what's called a linear operator. |
5DUQ3-Y_gX4 | So basically, the
change in the output df |
5DUQ3-Y_gX4 | is going to be a
linear operator. |
5DUQ3-Y_gX4 | Let me write that
as times, which |
5DUQ3-Y_gX4 | I'm going to call f prime of x
times the change in the input. |
5DUQ3-Y_gX4 | This is our-- whoops. |
5DUQ3-Y_gX4 | Actually, but my input
is in red, right? |
5DUQ3-Y_gX4 | That's my color code. |
5DUQ3-Y_gX4 | My color scheme is input
is red and output is blue. |
5DUQ3-Y_gX4 | So this is our dx. |
5DUQ3-Y_gX4 | So I'm going to interpret
this more generally. |
5DUQ3-Y_gX4 | If x is going to be some kind
of vector or matrix or whatever, |
5DUQ3-Y_gX4 | this is just going to be a
linear operation on this. |
5DUQ3-Y_gX4 | And of course, for numbers,
a linear operation, |
5DUQ3-Y_gX4 | this is just a number. |
5DUQ3-Y_gX4 | If dx is a number, the only
linear operation you can do |
5DUQ3-Y_gX4 | is multiply by a number. |
5DUQ3-Y_gX4 | So let me just remind you if
you haven't taken linear algebra |
5DUQ3-Y_gX4 | for a while, a review. |
5DUQ3-Y_gX4 | Let's talk about what
a linear operator is. |
5DUQ3-Y_gX4 | So suppose we have
some given vectors, v, |
5DUQ3-Y_gX4 | in some vector space. |
5DUQ3-Y_gX4 | That would be capital
V. And remember, |
5DUQ3-Y_gX4 | a vector space is anything
where you can basically |
5DUQ3-Y_gX4 | add, subtract, and
multiply by scalars. |
5DUQ3-Y_gX4 | We have a plus or minus
and times scalar operations |
5DUQ3-Y_gX4 | that stay in our vector space. |
5DUQ3-Y_gX4 | That's what the informal
definition of a vector space |
5DUQ3-Y_gX4 | is. |
5DUQ3-Y_gX4 | You can write out
axioms and so forth, |
5DUQ3-Y_gX4 | but that's basically
what it means. |
5DUQ3-Y_gX4 | A linear operator,
this is what we're |
5DUQ3-Y_gX4 | going to mean by linearization
in the derivative. |
5DUQ3-Y_gX4 | It has to be linear. |
5DUQ3-Y_gX4 | What does it mean to
be linear in general? |
5DUQ3-Y_gX4 | So we're going to call this-- |
5DUQ3-Y_gX4 | I'm going to denote this
by, let's say, L of-- |
5DUQ3-Y_gX4 | let me denote it by square
brackets or just by Lv. |
5DUQ3-Y_gX4 | When it's clear enough,
I'll just write it |
5DUQ3-Y_gX4 | as if it were a multiplication. |
5DUQ3-Y_gX4 | Often, that'll be clear enough. |
5DUQ3-Y_gX4 | This is really acting on-- |
5DUQ3-Y_gX4 | when I write Lv,
it's not necessarily |
5DUQ3-Y_gX4 | an ordinary multiplication. |
5DUQ3-Y_gX4 | This is just going
to be acting on v. |
5DUQ3-Y_gX4 | So a linear operator is a rule
that basically takes a vector |
5DUQ3-Y_gX4 | and gives you a vector out maybe
in a different vector space. |
5DUQ3-Y_gX4 | So this is L takes
a vector in, v in, |
5DUQ3-Y_gX4 | and it gives you a vector-- |
5DUQ3-Y_gX4 | that's a terrible L-- |
5DUQ3-Y_gX4 | Lv out maybe in a
different vector space. |
5DUQ3-Y_gX4 | And linearity means
what you think it means. |
5DUQ3-Y_gX4 | It means if you
take, for example, |
5DUQ3-Y_gX4 | L of v1 plus v2, if you take
the sum of the inputs, that's |
5DUQ3-Y_gX4 | the same thing as L
of v1 plus L of v2. |
5DUQ3-Y_gX4 | So if you add inputs,
that's the same thing |
5DUQ3-Y_gX4 | as adding the outputs or if
you multiply by a scalar. |
5DUQ3-Y_gX4 | And so as usual
in linear algebra, |
5DUQ3-Y_gX4 | Greek letters are going to
denote scalars as you would. |
5DUQ3-Y_gX4 | So that's equal to-- |
5DUQ3-Y_gX4 | you can pull out the scalar. |
5DUQ3-Y_gX4 | And so the nice thing
about linear operators |
5DUQ3-Y_gX4 | is we can define them on lots
of kinds of vector spaces. |
5DUQ3-Y_gX4 | So let's just do a
couple of examples |
5DUQ3-Y_gX4 | just to make sure we're
on the same page here. |
5DUQ3-Y_gX4 | And so, for example,
you could just |
5DUQ3-Y_gX4 | have L is multiplication
by a scalar. |
5DUQ3-Y_gX4 | So you can have just L
of v is just alpha v. |
5DUQ3-Y_gX4 | That's a perfectly
good linear operation. |
5DUQ3-Y_gX4 | And if your vector space,
if your vs are scalars, |
5DUQ3-Y_gX4 | this is the only option. |
5DUQ3-Y_gX4 | If vs are, say,
real numbers, that's |
5DUQ3-Y_gX4 | a perfectly good vector space. |
5DUQ3-Y_gX4 | Another one that you're
very familiar with |
5DUQ3-Y_gX4 | is if L is a
multiplication by a matrix. |
5DUQ3-Y_gX4 | ALAN EDELMAN: Steven, maybe I'll
just point out sometimes people |
5DUQ3-Y_gX4 | like to ask me. |
5DUQ3-Y_gX4 | Wait, I thought that the linear
operators on scalars are, |
5DUQ3-Y_gX4 | I think in high school notation,
y equals mx plus b, right? |
5DUQ3-Y_gX4 | It's scalar times-- |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Yeah, yeah. |
5DUQ3-Y_gX4 | ALAN EDELMAN: --plus an
offset that may not be 0. |
5DUQ3-Y_gX4 | So what's going on here? |
5DUQ3-Y_gX4 | Is that linear or not linear? |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Yeah. |
5DUQ3-Y_gX4 | So what about-- yeah,
so let's do that. |
5DUQ3-Y_gX4 | Let me call it a
different thing. |
5DUQ3-Y_gX4 | What letter should I use? |
5DUQ3-Y_gX4 | O, let's use O. Ov equals 2v
plus 1 for v is real numbers. |
5DUQ3-Y_gX4 | ALAN EDELMAN: Is that
linear or not linear? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.