WEBVTT
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ok so last time we will talk a little bit
about a linear transformations now what i
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want to do this time specifically is focus
on linear transformations and what they do
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to areas ok so for instance so so recall the
following features we said linear transformations
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being a special case of the fine transformations
to the following they map lines to lines ok
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so this is a so what all do we know linear
transformations certainly map lines to lines
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well are points as we mention that could maps
some lines just to a single point
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and what are just you do well what is it do
to lengths what is it do to angles
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now answers to these questions at least we
know through the various examples that we
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have looked at before so there are instances
of linear transformations which preserve lengths
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and there are others which might in a uniformly
increased length by some factor and get others
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which might do need there of the you know
along some directions increased length by
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some factor along a different direction they
may increased length by some other factor
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and so on so as for as lengths are concerned
well there is nothing you know uniform that
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you can really say ok so can't say much really
about lengths in general if i don't know any
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extra information about the linear transformation
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similarly angles so again we looked at instances
of transformation which preserve angles such
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as rotations reflections dilations and so
on but then there were instances is short
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of the inhomogeneous dilation for instance
which did not preserve angles so again we
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can't really say much ok so a typical linear
transformation my deform the plain in such
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a way that neighbor lengths nor angles or
preserve ok so it does trains things to the
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space now areas is sort of the the third aspect
that we we studied in our various examples
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so what's that the difficult thing we want
to do let's do the following lets pick some
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region in the plain so for instance i could
take so just make life simpler let's take
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a polygonal region ok i which i mean uh let
let say i take region bounded by lines
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a need not be a regular polygonal so i could
take write could take so i something like
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this
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so here is the region are that a write down
on the plan i am allowed to choose various
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possibilities i could take may be a triangle
region or a square rectangle pretty much any
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any sort of polygonal region that you can
think of and what i want to do is to study
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the effect of uh applying this function f
so let me say that i have a function f for
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linear transformation so let f be a linear
transformation ok we could in fact also allow
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a fine transformation but let's just strict
to linear for now i want to look at what f
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does to this region r ok so the first thing
is in all as we set before its sort of enough
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to really figure out what happens to each
of these various vertices so what f would
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do is to map these vertices to some points
so there would be could say one two three
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four five so maybe i have again five distinct
vertices
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so maybe it it deforms set in some other fashion
right so it may not have the same form ask
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the original region so this is my region r
and maybe that's my region r dash ok and r
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dash is just the image of r under my under
this map ok so i am just sort of depict in
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some things schematically here it's probably
not very good picture in general so
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now here's the question i want to compare
the area that's enclose by the region r with
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the area that's enclose by the region r dash
right so this is the difficult thing that
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we have looked at earlier by what factor does
thus the area increase or decrease so here's
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the question i want area of the region r dash
so r dash is what we would call f of r that
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is the image of r under the function f so
called r dash divided by the area of r
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so this you might want to call the the area
scaling factor or the area dilation factor
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now he is the surprising fact if f is the
linear transformation then this ratio of is
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a constant independent of what shape r has
are where it is located on the plain ok so
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surprising fact this is f is a linear transformation
are in fact also an a fine transformation
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then this area scaling factor is a constant
by constant i mean it does not depend on its
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independent of the region r it does in matter
what shape the region has where it is located
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and so on and so for
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so it's absolutely independent of r the answer
will always be the same ok and so what is
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it mean in other words f is f dilates area
uniformly by this this constant ok so let's
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give this constant a name let's call it say
delta this is a constant so in other words
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i e f dilates areas uniformly by this factor
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ok so all areas expand by the same amount
that's basically what this moves even though
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so what why did i call this somewhat surprising
because lengths for instance could suffer
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you know different amounts of dilation and
different directions and so on but when you
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talk about areas what you get is the same
uniform dilation ok
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so lets us do this again by example let's
take
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f to be twelve i guess i am just taking most
general example f of xy let's i said it's
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a linear transformation so it something which
looks like this right ax plus by cx plus dy
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and let see what this dilation factor area
dilation factor delta could be so in order
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to do this let's take a simple figure r let's
just take to unit square so i will take the
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origin one zero zero one one one thing of
this has being my region r so r of course
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has area one in this case it's just a square
of side one and i want to apply this function
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f to it now observe that applying this function
f
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does the following it maps this square too
well in general parallelogram so here's
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what in square maps to maps to a parallelogram
ok so how how do they deduce this well i will
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just figure out the four end points so for
instance the origin maps to the origin i just
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plug in zero zero one comma zero maps to the
point a comma c by plugging in this maps to
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the point bd and the point one comma one maps
to well a plus b comma c plus d so these four
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are in fact vertices of a parallelogram ok
and so now the question really is in order
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to figure out this dilation factor so let's
as work out so if for the roman let's accept
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this fact has being true that this dilation
factor is is uniform that just a single dilation
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factor delta so to figure out what delta is
all we have the do is take this particular
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choice of r and r dash
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so i compute the area of the parallelogram
and divided by the area of those certain so
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let's compute this scaling factor delta is
just the area of the parallelogram divided
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by the area of the original region r now of
course the original region has area one so
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i don't need do anything there following into
figure out this the area of the parallelogram
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ok so of course the area of the parallelogram
uh there are several different ways of trying
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to compute the area of a parallelogram but
uh he is one so if you have a parallelogram
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in general so let me just say brief due to
talk about areas and parallelograms and formulas
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were those if i have say this theta the angle
now the the usual formula says it is uh base
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times altitude
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so let's may be call these two sides as something
uh we will this p and q are the two sides
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the area of a parallelogram is just base times
so p times the altitude so area equals length
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of the base times altitude so bases just p
and the altitude so just by elementary trigonometry
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is q sin theta ok so q sign theta is this
vertical line segment here and so it is p
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times q times sin theta now [vocalized-noise]
of course sin theta so the angle uh here could
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be uh you it depends on what the angular something
between zero and an hundred and eighty degrees
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now uh what do we know about this quantity
here pq is sign theta we could try and compute
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it various space but the easiest way to do
this is just by using you um going back to
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uh definition of vector cross products
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so recall so we have already talk to about
cross products in one of the earlier lectures
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so recall if a and so i will just use notation
for vectors if vector a vector b are two vectors
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then there cross product a cross b has the
following magnitude the length of the cross
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products is exactly the length of a times
the length of b times sin of the angle between
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them ok so i am i am actually uh sort of appealing
to some previous knowledge of cross products
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so this just make set much easier so if you
are familiar with this its quicker to to get
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to the area formula
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if not there are sort of other way is that
you must a player on with in order to to get
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to the same answer that i will get to in a
moment so what this as is to find the area
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of a parallelogram you just thing of the two
sides has been vectors so you think of these
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as a vector this as a vector and the area
is just the magnitude of the cross product
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ok so that's a very nice description of the
area so let says apply this to this situation
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so let's come back here we will trying to
figure out the area of this parallelogram
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or prime so what we want to do is to think
of one side and the other side of both being
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vector so this is nothing but the absolute
value of the cross product of these two vectors
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ok and now i will use vector notation again
so recall that i cap and j cap are often the
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standard notation for the unit vectors along
the the x direction and the y direction
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so i have this vector is ai cap plus cj cap
cross product with bi cap plus dj cap so
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lets a cross product of these two vectors
and i need to find the magnitude of the cross
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product and so here what we need is the definition
of the cross product so [vocalized-noise]
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if you actually just compute this cross product
here's what will you get it is going to be
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so this is the magnitude of so for instance
i cross i is zero so you should just think
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of this as unit expand this out completely
distributing it using the distributive law
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i cross i is zero i cross j is k so this just
gives you ad times k and then the other term
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there we just give minus bc finds k ok and
k cap is just the unit vector along the the
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the z axis and so this final answer here is
just the absolute value of ad minus bc
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ok so what is this mean val this just says
that this dilation factor so here's the conclusion
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the dilation area dilation factor delta is
nothing but the absolute value of ad minus
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bc ok and now what are a b c and d recall
those were just what appeared in the definition
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of the function the linear transformation
is exactly ax plus by cx plus dy so the a
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b c and d are these four numbers right and
recall again that these are best encoded in
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the form of a two cross two matrix so should
really thing of this linear transformation
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as being encoded by this matrix a b c d and
now that we do this the number ad minus bc
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again has another interpretation so ad minus
bc recall is just what we will call the determinant
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of this matrix
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so recall that the determinant of this two
cross matrix here is exactly ad minus bc ok
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so what this means is that the scaling factor
has a natural interpretation in terms of the
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matrix of the linear transformation so thus
conclude that the scaling factor delta is
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nothing but the absolute value of the determinant
of the matrix earlier transformation is the
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final conclusion ok so in general for an arbitrary
linear transformation for given the transformation
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f itself and you want to know by what factor
does its scale areas all you want to do is
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to just write out the matrix is corresponding
to the linear transformation and then compute
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its determinant ok and the absolute value
of the determinant is exactly the thing that
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we want the scaling factor ok
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so uh returning in some sense to something
it started out with let see is do the something
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for functions from r two r ok so let's do
the following let's consider let's go one
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dimension lower instead of considering functions
from r two to r two let's consider functions
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from r one to r one so by which we mean just
functions from the real line to itself and
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here what would a linear transformation mean
well a linear transformation from r one to
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r one just copying the definition that we
used for two variables a linear transformation
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now can only act on one variable and only
give you one real answer so i mean the run
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out two components valid just something of
the following form it just ax if i have two
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variables i can do ax plus by cx plus dy and
so on if i only have one variable x all i
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can do just multiply by some constant ok
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and such a thing is what you would caller
linear transformation in one variable so a
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here is some constant ok so such a thing is
linear transformation now well so again you
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know you could try and do the same notions
that we had earlier for instance we can now
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talk not about area dilations but of length
dilations so you could now ask the following
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things since we are now in just one dimension
the notion of area is most naturally replace
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by lengths so for instance you can ask suppose
i took the line segment of some length l and
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i apply the function f to it what it be the
new length of my line segment ok so for instances
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let say i start imagine the line segments
starts at zero and goes to l and now we apply
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this function f to it and ask what happens
to this line segments
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so let me call this interval let me call it
i is no interval i apply my function f of
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zero is zero and of course by definition f
of l is just the point a l ok so this interval
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i when i apply the function f just becomes
potentially a lager interval so if a is for
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instance two it's an interval of size two
ok so this is my new interval i dash and so
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now you can ask what is the length dilation
factor in this case ok so here the length
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dilation factor for the the map f is the following
it's just the length of i dash
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divided by the length of i and observe here
that i dash has a times the length of i
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i this as this is al and this is a so this
just going to be a ok so this is of course
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if a is positive if a turned out to be a negative
number then what you would get would really
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be the one of the same interval but in the
opposite direction case of the length in that
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case should really be the modulus of al so
actually speaking if a is negative since you
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know the both top and bottom are positive
numbers what i should get get out as the answers
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is a positive number so here's the general
answer the length dilation factor for max
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from r one to r one is just the absolute value
of a just the absolute value of the the constant
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ok now observe again they just like linear
transformations from r two to r two here the
19:49.100 --> 19:56.009
length dilation is uniform pretty much no
matter value keep this uh interval of length
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l no matter where you place it on the real
line it will always be expanded by the same
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ratio a ok
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by the same factor a so this what we mean
by uniform length dilation it always expands
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by the same factor independent of where its
placed similar in the case r two no matter
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what the shape of your region and no matter
where it placed it always expands by that
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same factor delta which is the data mine so
now uh let's just makes the so little bit
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more general so here's one final point about
general functions from not linear transformations
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we just take and arbitrary functions from
r two r ok so let's say it's a nice enough
20:43.100 --> 20:47.460
functions a maybe it is a continuous function
it's a differentiable function things like
20:47.460 --> 20:53.919
that so imagine and nice smooth graph of
this function so if i have a saying arbitrary
20:53.919 --> 21:02.059
function sufficiently nice and arbitrary for
now uh let's ask for this same same dilation
21:02.059 --> 21:03.619
factor designs
21:03.619 --> 21:09.100
so here's what it means so i have this function
f its not linear necessarily what i want to
21:09.100 --> 21:16.809
do is following i want to take a point x not
on the real line i want to take and interval
21:16.809 --> 21:24.080
so i want to take say in interval i one of
points and one of its end points x not ok
21:24.080 --> 21:31.730
so i take x not and x not plus so the other
end point let's called it delta x ok for now
21:31.730 --> 21:35.859
delta x is just any real number but eventually
will thing of it as being a very small real
21:35.859 --> 21:41.590
number so other these are very small interval
so i pick some interval around x not are with
21:41.590 --> 21:47.330
x not is one of its end points and ask well
what happens to this interval when i map it
21:47.330 --> 21:52.240
under the function f ok so what happens to
the two end points for instance x not will
21:52.240 --> 22:01.499
map to the point f of x not the right end
point will map to say this right end point
22:01.499 --> 22:06.889
x not plus delta x ok
22:06.889 --> 22:16.450
so the new interval i get is exactly this
gui i dash so i can ask for the same question
22:16.450 --> 22:31.159
what's the dilation factor what's the length
dilation ok from comparing i with i dash well
22:31.159 --> 22:39.750
this is the length of i dash the length of
i dash divided by the length of i and that's
22:39.750 --> 22:51.330
exactly well what's the length of i dash it
is f of x not plus delta x minus f of x not
22:51.330 --> 22:55.950
well actually i should put modulus because
i don't know f of x not plus delta x could
22:55.950 --> 23:00.320
be to the left of f of x not in my diagram
i drawn it to the right but i don't know which
23:00.320 --> 23:06.919
way it it lies so it's the absolute value
of this difference divided by divided by the
23:06.919 --> 23:11.059
length of i the length of i is exactly delta
x ok
23:11.059 --> 23:20.960
so this is just sorry delta x is any way positive
so i can as put it within the modulus
23:20.960 --> 23:27.259
so here's the answer the length dilation factor
comparing i with i dash so this is what we
23:27.259 --> 23:33.299
would sometimes called it's the length dilation
factor act the point x not right observe if
23:33.299 --> 23:38.090
i change i to lie somewhere else then of course
the length dilation factor we had to take
23:38.090 --> 23:45.379
those two uh end points in so the length dilation
factor it x not is really this quotient here
23:45.379 --> 23:51.200
and now i am think of delta x as being uh
smaller and smaller number as shall we now
23:51.200 --> 23:59.570
let delta x also approach zero let it go to
zero so when you do this what you doing really
23:59.570 --> 24:03.789
is taking smaller and smaller and smaller
intervals around x not
24:03.789 --> 24:11.729
and asking by what factor is there lengths
by related by this function f ok and so we
24:11.729 --> 24:17.509
want to really consider the limit of the right
hand side as delta x goes to zero and that
24:17.509 --> 24:24.019
limit is really well it it should be familiar
if you've seen this before is just the derivative
24:24.019 --> 24:40.359
the limit has delta x goes to zero of this
quotient
24:40.359 --> 24:44.480
so this quotient here is of course just the
derivative so assuming the derivative at x
24:44.480 --> 24:49.929
not exist but this when it exist what you
have is just the absolute value of the derivative
24:49.929 --> 24:56.960
at that point ok so what this analysis terms
as as the following geometrical interpretation
24:56.960 --> 25:03.700
of the derivative the absolute value of the
derivative essentially tells you the dilation
25:03.700 --> 25:05.950
factor at that point ok
25:05.950 --> 25:22.570
so what is this this is exactly the dilation
length dilation factor of the function f at
25:22.570 --> 25:31.859
the point x not ok so all it's doing is just
keeping track of get the amount by which intervals
25:31.859 --> 25:38.190
gets scaled when those intervals are in a
small neighborhood of x not ok but observe
25:38.190 --> 25:44.289
that un like the case of a linear transformation
this is not uniform the derivate of the value
25:44.289 --> 25:48.899
of the derivative may not be a constant at
all points right it is of course a constant
25:48.899 --> 25:56.129
if f is a linear transformation so observe
if f of x is just the function ax then we
25:56.129 --> 26:01.279
said that this the length dilation factor
so if you compute f prime are at x not for
26:01.279 --> 26:02.669
any point x not
26:02.669 --> 26:08.159
the answers always a so the absolute value
is just the absolute value of a ok this is
26:08.159 --> 26:19.429
independent of x not this for all x not that's
the reason why if it is a linear transformation
26:19.429 --> 26:24.309
you always get a uniform dilation that uh
the amount of dilation is always modulus of
26:24.309 --> 26:29.190
a no matter what point x not you are talking
about but for the general function for the
26:29.190 --> 26:34.200
general differentiable function this may not
happen for different points x not the value
26:34.200 --> 26:40.279
of the derivative might be different it and
so what the function really does is to dilate
26:40.279 --> 26:44.700
intervals differently depending on where the
pointers so this is really and non uniform
26:44.700 --> 26:50.799
dilation in general ok but nevertheless this
this just particular geometrical interpretation
26:50.799 --> 26:58.190
is useful to keep in mind and this is sparkly
why the we talk to about area dilations
26:58.190 --> 27:02.340
for example in the case of uh maps from r
two to r two
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the more general thing there so so the natural
question here is what if we were not considering
27:08.400 --> 27:13.649
linear transformations from r two to r two
what if we had a more general function from
27:13.649 --> 27:18.359
r two to r two say it differentiable lines
on so then it turns out that you can still
27:18.359 --> 27:23.669
try and figure out what the area dilation
factor would be and that would now involve
27:23.669 --> 27:28.679
the notion of partial derivatives ok so that's
the reason are that's one natural way of thinking
27:28.679 --> 27:33.610
about what partial derivatives do uh they
really give you a way of trying to compute
27:33.610 --> 27:39.450
the area dilation factor at each factor ok
so will talk to little bit more about all
27:39.450 --> 28:01.649
this next time