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so today we are going to start looking at
another grand generalization from school geometry
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or three dimensional geometry this is inner
product spaces or hilbert spaces as i said
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the nice property of three dimensional world
or the geometry that we study equivalent geometry
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that we study is orthogonality between two
vectors if we have perpendicular vectors we
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can define models very conveniently we can
define vectors very conveniently
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so there are many many advantages of pythagoras
theorem and we would like it to hold in a
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general space which consist of functions which
consists of polynomials and so so we have
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to come up with new structure on a vector
space which is equivalent to what we have
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available in three dimensions and then tried
to see to it that the properties that are
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importance in three dimensions or in school
geometry are also preserved in these newly
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defined vector spaces
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so now we impose one more structure see till
now we started by defining norms but just
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length or norm was not enough it was helpful
in defining generalizing the ideas of convergence
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convergence to a limit and so on but we need
something more we need angles so remember
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two things what i want to generalize i want
to generalize
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the so called dot product if my x is nothing
by r3 then if i am given any two vectors x
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and y that belong to x then what i do is i
construct unit vector as x by 2 norm of x
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and y cap which is also unit vector y by two
norm of y so this is two norm x two norm is
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x transpose x raise to half we construct two
vectors which are unit vectors in direction
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of x and y
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and then if i want to find out angle between
x and y i have to take a dot product between
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so fundamental result that we have is cos
theta equal to x cap transpose y cap a fundamental
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result that i have is x cap transpose y cap
equal to cos theta angle between them i want
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this particular idea to be generalized in
a vector space now what we know from trigonometry
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is that cos theta is always bounded between
plus or minus 1
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so another way of stating this equality is
to say that mor cos theta is less than or
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equal to 1 or this also means that in three
dimensions x by norm x this is another way
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of writing the same inequality cos theta is
always less than 1 so x transpose so this
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is a scalar and this is also a scalar so i
can write this as mod of x transpose y is
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always less than or equal to
and then what was important property of that
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we said we want to have is orthogonality
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so when x is perpendicular to y x transpose
y equal to 0 this was very very important
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for us we extensively used orthogonality we
use orthogonal basis for example the most
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well known orthogonal basis is i j k unit
vectors perpendicular along the coordinate
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directions so orthogonality is a very very
important property we want it so i want to
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now come up with generalization of these results
in a vector space that is my aim so i am going
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to define a new entity called inner product
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i am given a vector space x together with
a function called inner product
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so i am given a vector space i am given a
vector space x and a function which is defined
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on x times x to the field f so given xy that
belong to x
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i am going to define inner product x inner
product y this is the notation that we are
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going to use throughout the course x inner
product y is defined from x into x to so what
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are the axioms that govern this definition
there are four axioms the certain properties
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which are generic to inner product in three
dimensions which i want to now generalize
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and come up with a generalized definition
which will in a special case would be this
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dot product which you know in three dimensions
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so my first axiom is well when i am working
with vector spaces in many situation i have
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to work with complex valued functions and
complex valued vectors so what this says is
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that if i change the order if i take inner
product of x with y then and if i change the
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order what i get is the complex conjugate
so typically the field that we are going to
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work with is rrc set of complex numbers
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so well if you are working with real valued
vectors or real valued functions then this
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is very obvious if i change the order if i
make x transpose or y or y transpose x i am
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going to get the same value with complex numbers
you get a complex conjugate that is important
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the second property i want the inner product
to observe is that if i given any three vectors
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any x y and z that belong to x
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so this inner product that we define should
distribute over vector addition so if i take
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x plus y and take inner product with z then
that is same as adding these two inner products
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x with z and y with z that is the second important
property of a function to qualify as an inner
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product so what is the third important property
the third important property is that if i
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take a scalar lambda and multiply with x then
this is same as lambda bar
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this is same as but the way it happens with
the second element and the first element is
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different in inner product
if you are working with real numbers these
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both results are same because lambda bar equal
to lambda so if you are working with complex
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numbers you need to separate these two equalities
so if i take the first vector into lambda
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that will be same thing as multiplying inner
product of x and y with lambda bar
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that is complex conjugate of lambda and if
i take the second vector multiplied this is
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y if i take and multiply by lambda and it
is same as lambda into so this is another
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essential property of this is an axiom that
defines a function to be inner product i want
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to maintain this because we have to generalize
a few things so i want to draw parallel so
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let this be there for sometime what is the
last axiom
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the last axiom is the fourth axiom is if i
take inner product with let us look at here
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let us look at this property if i take inner
product of a vector x with itself what do
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i get i get two norm in three dimensions what
is two norm two norm is if x1 x2 x3 are three
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components x1 square plus x2 square plus x3
square whole to the power half so this particular
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property is quite important in light of generalizing
this
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so this should always be greater than or equal
to 0 and
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the inner product of x with itself should
be 0 only if x is 0 this is also very very
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important here in three dimensions only inner
product of x will be equal to 0 x transpose
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x will be 0 only if it is origin the same
property is being generalized here in fact
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this is what helps us in defining a norm which
is tied up with inner product a norm which
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is tied with inner product
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a norm which is tied up with inner product
plays very very important role in numerical
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analysis because this is a norm which comes
with a definition of angle that is why two
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norm is something which is very very often
used in applied mathematics so now let us
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start looking at can we define there in three
dimensions we defined a norm using inner product
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can i do it here in a general vector space
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so i said any function that obeys these four
axioms qualifies to be an inner product so
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it is not necessary that we have to you know
we have only one particular way of defining
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inner product we have a generic way of coming
up with a definition of inner product that
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is suitable to our application what i mean
by this will become clear as we go along so
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let me define some examples of inner products
which are even on r3 i will show you that
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there are different ways of defining inner
products on three dimensions
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but before that let me just state what is
the hilbert space sometime back in the last
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lecture you heard about banach spaces “professor
- student conversation starts” what are
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banach spaces complete norm linear spaces
so what happens in complete spaces every cauchy
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sequence is convergent in the space “professor
- student conversation ends” so hilbert
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space
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a complete inner product space is called as
a hilbert space this name is given after a
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great mathematician hilbert who laid foundations
of functional analysis so now some examples
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of inner products
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my first example is to show that there is
no unique way of defining inner product in
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three dimensions also let me take my x as
r3 now when you say r3 the field is r i am
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not going to write it every time this to keep
the things simple i am going to define the
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inner product on this now which is different
from what we have done earlier you know x
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transpose x
so let w be a positive definite matrix
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so now we define an inner product using this
positive definite matrix w so my inner product
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for any two vectors x y that belong to r3
so i have this x and y belong to r3 and x
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inner product y i am going to give a little
subscript here w is going to be defined as
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x transpose w y
a simplest example of w would be a diagonal
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matrix see for example simplest example of
w would be you know matrix which is 100.1
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and 1000
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now you might ask me why do you want to define
some funny matrix w and then call it as inner
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product where is it useful that is why i am
working with a reactor and my x is a vector
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that consist of say temperature pressure and
concentration fractional so this is in 10s
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and 20s it is a degree centigrade temperature
pressure let say is you know defining pascals
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mega pascals so it is in 10 to the power 5
something here you know x is in fractions
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if i use my old good old way of defining inner
product or length i have trouble because this
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mod fraction is always going to be a small
number square of it is going to be smaller
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number so many times i need to work with scaled
variables i need to work with scaled variables
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at such times it is useful to have an inner
product definition which normalizes the unique
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differences between different variables
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i am not just defining this w matrix just
like that there is purpose behind this under
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many many situations value you will get into
this kind of normalization business where
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you have to use a matrix w now let us see
whether this particular inner product satisfies
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properties that are specified what is the
first property just go back and look at your
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so first property is my first property is
that x y should be y x bar but we are not
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working with complex numbers we are not working
with complex numbers since we are not working
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with complex numbers in this particular case
it is if i interchange it should not matter
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so this i do not have to prove to you that
x transpose wy is same as y transpose wx why
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this is true
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this is always true i need one more property
i have missed out something here “professor
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- student conversation starts” no i just
said it is a positive definite matrix i need
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something more yes so i need this matrix w
with positive definite and also this w has
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to be symmetric w equal to w transpose
this should be symmetric otherwise this does
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not hold otherwise this does not hold
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“professor - student conversation ends”
so this matrix being positive if it is not
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sufficient it should be a symmetric positive
definite matrix then what will happen if i
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take x transpose wy transpose that is y transpose
w transpose x which is y transpose wx because
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w transpose equal to w so symmetry is very
very important symmetry is very very important
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so i need a positive definite matrix and a
symmetric matrix
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what next lambda into x what will happen what
is the next property i think x plus y the
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second property distribution is very obvious
we do not have to prove this x plus y z equal
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to x plus y transpose wz which is nothing
but xz plus yz i think this is just it just
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follows very simple what is the third thing
if i multiply one of the vectors by a scalar
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inner product should get multiplied by now
we do not have to do mod here there is no
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bar here we just have to take the scalar out
because we are working with real number so
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the third property is very very obvious
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that is lambda into xy is lambda x transpose
wy which is lambda x transpose wy i do not
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want a complex conjugate because we are working
with real numbers what about the fourth property
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does it hold x transpose wx if take inner
product of a vector with itself what is the
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meaning of positive definiteness all the eigen
values are greater than 0 there is no zero
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eigen values all the eigen values are greater
than 0
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the definition of positive definiteness itself
means
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this is the definition of positive definiteness
a matrix is positive definite the fundamental
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definition of positive definiteness is that
if x transpose wx is always greater than 0
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if x is not equal to 0 if x equal to 0 it
will be equal to zero so only vector that
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will give you x transpose wx equal to 0 is
0 vector that is what we wanted all four examples
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are satisfied
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so this is another way of defining inner product
on three dimensions these kind of inner product
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be very very routinely used in numerical methods
because we need to do scaling of variables
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x will consist of pressures temperature concentrations
all kinds of variables which have different
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units and then if you want to find out length
of such a vector you cannot just say x1 square
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plus x2 square plus x3 square
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you need to multiply by a suitable waiting
matrix that is why you need this “professor
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- student conversation starts” that is why
i said w has to be positive definite and symmetric
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symmetric is important x transpose wy will
be symmetrical so just positive definiteness
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is not enough we need symmetry also so symmetric
positive definite matrix is important and
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then sir what is lambda bar lambda bar is
complex conjugate but we are working right
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now with real numbers so complex conjugate
will be real number itself “professor - student
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conversation ends”
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so this i can very easily change to my second
example where you talk with rn i could have
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talked with rn and the same thing would hold
i have a symmetric positive definite matrix
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and i can define a norm which is i can define
an inner product which is using any symmetric
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positive definite matrix which is n into n
which will give me all the properties that
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are need for defining inner product we still
have not established the connection between
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the last axiom and the norm
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i have been just saying that well it is related
to the inner product inner product gives you
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a norm which is here but actually we need
to see that connection so i will give one
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or two examples and move to proving that actually
inner product in a general space defines a
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norm just like in three dimensions x transpose
x gives you a norm
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you will also get a norm defined through inner
product before doing that let me give you
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one or two more examples of inner product
spaces so my second example would be rn or
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i can easily move to cn a complex valued and
so on where the matrix there should be hermitian
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not symmetric positive it should be hermitian
moving on from finite dimensional spaces let
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me give you third example
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so set of square integrable functions over
an interval ab set of square integrable functions
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over the interval ab you have come across
this kind of a set when you worked with fourier
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series expansion now you will soon realize
what are the connections so if i am given
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any two functions say ft and gt that belong
to x then i can define an inner product between
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ft and gt as integral a to b set of all square
integral functions
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so integral over a to b typically when you
study fourier series in your undergraduate
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we look at a b that corresponds to 0 to 2
pi or we look at ab that correspond to - pi
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to pi you remember something like this when
you do fourier series expansion you take sin
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theta or sin t * ft dt integral sign t ftdt
that is actually inner product and you can
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just check whether all four axioms are satisfied
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let us look at first axiom what is the first
axiom if i interchange f and g will the integral
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be different so first axiom is satisfied if
i multiply ft by some lambda what will happen
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to the integral it will be lambda times second
property is satisfied what about distribution
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if i take f plus g inner product with some
ht it is very obvious the third property
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if i take ft plus gt inner product with xt
this will be integral a to b ft plus gt which
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is same as integral a to b
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everyone with me on this so the third axiom
is satisfied what about the fourth axiom if
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00:31:06,600 --> 00:31:12,360
i take inner product of a function f with
itself what will happen will always be a positive
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number why
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so my fourth axiom is integral of ft with
ft this is nothing but integral a to b ft
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square dt which is always greater than 0 if
ft is not a zero function
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am i correct if ft has even one nonzero value
in interval a to be ft square will be positive
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ft square dt will be positive so ft as long
as this will be zero when f is zero every
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where on a b if f has nonzero values this
integral will always be nonzero
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so all the four properties that you need for
an inner product space or inner product to
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be defined are satisfied i could further modify
this inner product see just like from x transpose
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x from x i said x transpose wx where w is
a symmetric positive definite matrix is also
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inner product i could expand this definition
by putting a positive waiting function here
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so i can have another definition my fourth
example would be
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i will take waiting function wt ft gt dt wt
is strictly greater than zero on wt is strictly
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greater than 0 is a positive function wt is
a positive function it has only positive values
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in the interval ab this is my interval ab
on which inner product is defined on which
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the space is defined just like you could use
the positive definite symmetric matrix there
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if i modify my definition of inner product
by multiplying a positive waiting function
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that also satisfies inner product
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and these kinds of waiting functions we are
going to hit up on soon in when we come up
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with different ways of defining inner product
on set of continuous functions which are square
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integrable we will also come up with these
kind of inner products we will need them when
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you solve partial differential equations so
boundary value problems when you solve in
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the mathematical methods scores
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so there are different ways of defining inner
products yet we have to establish two major
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connections one is with the angle and other
is with the norm so let me start preparing
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for this i need to prove an inequality which
is essence which exactly captures this part
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in order to show that an inner product defines
a norm i need to pull an inequality called
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as cauchy schwartz inequality
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and this inequality will help us to come up
with connection between inner product and
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the so called two norm this is two norm and
we want a connection to be established in
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a general
so
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what all things that you need for a function
to be a norm when do you call a function to
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be a norm what are the three axioms
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one is norm x greater than 0 if x not equal
to 0 and this is equal to 0 if x equal to
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0 that is the first axiom what is my candidate
norm definition is i want to use in an inner
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product space i want to define a norm actually
we will call it two norm but right now let
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us keep calling it two norm here is i want
to say x x raise to half that is what i want
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to do is x transpose x this is my candidate
function
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now does this follow the first axiom for norm
does it follow from the definition of the
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very definition it will follow nothing to
worry what is the second thing about norm
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scalar multiplication so if i take alpha into
x then that is equal to mod alpha norm x what
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00:36:54,040 --> 00:37:10,670
about this
does it follow let me see alpha x alpha x
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00:37:10,670 --> 00:37:23,620
what is this equal to
alpha bar alpha x x right first element alpha
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00:37:23,620 --> 00:37:36,780
bar second element alpha so which is mod alpha
square x x
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00:37:36,780 --> 00:37:53,810
so alpha x alpha x raise to half equal to
mod alpha x x raise to half we have proved
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whatever we wanted so far so good
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now comes the third problem what is the third
thing triangle inequality triangle inequality
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is where we need this to be generalized you
cannot go to triangle inequality unless you
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generalize this result in inner product spaces
and here we need a little bit of work i am
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going to prove this on the board why this
defines a inner product why this inner product
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defines a norm and how you can generalize
this result in an inner product space is called
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00:38:42,580 --> 00:38:57,510
as cauchy schwartz inequality
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so that what is that i want to do i want to
generalize this particular result from three
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dimensions except here it is written x transpose
y i want to prove i want to arrive at mod
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00:39:17,310 --> 00:39:29,600
of xy is less than or equal to norm x if i
take inner product of any two vectors xy is
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00:39:29,600 --> 00:39:40,650
absolute value is less than this is what i
want to show in any inner product space that
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00:39:40,650 --> 00:39:46,100
is generalization of the result cos theta
is less than 1 and with that i will move to
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00:39:46,100 --> 00:39:52,730
triangle inequality because i have to establish
triangle inequality to come up with
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00:39:52,730 --> 00:40:04,730
so how do i do that now let us first look
at the situation were y equal to zero vector
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if y equal to 0 vector does this hold always
because in a product with zero will give you
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00:40:12,030 --> 00:40:30,600
zero 0 less than or equal to 0 so if y equal
to 0 so we do not want to look at a trivial
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00:40:30,600 --> 00:40:44,770
case zero vector case now to prove this inequality
now i am going to play a trick so let lambda
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be a scalar nonzero scalar such that
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i am going to take a vector x minus lambda
y and take inner product of x minus lambda
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00:41:11,280 --> 00:41:14,510
y with itself everyone with me on this
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00:41:14,510 --> 00:41:26,450
lambda is any arbitrary scalar so does this
hold for any nonzero lambda this inequality
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00:41:26,450 --> 00:41:33,109
holds for in why inner product of a vector
with itself is always greater than or equal
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00:41:33,109 --> 00:41:54,450
to 0 so this always holds for any lambda not
equal to 0 it holds for any lambda so what
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00:41:54,450 --> 00:42:00,560
is this quantity on the left hand side can
you expand this so this will be x inner product
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x i am taking first with first when x inner
product lambda x y minus lambda bar y x look
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00:42:19,270 --> 00:42:25,200
carefully lambda times lambda and x distribution
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i am using the distribution property plus
everyone with me on this i have just expanded
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00:42:44,890 --> 00:42:51,240
the right hand side this also of course has
to be greater than 0 this is x inner product
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00:42:51,240 --> 00:43:00,070
x always greater than 0 mod lambda square
y always greater than 0 now i have two quantities
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00:43:00,070 --> 00:43:17,210
in between x and y and y in the product x
so let us preserve this part here because
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00:43:17,210 --> 00:43:25,840
this is what we are generalizing so this holds
for any lambda am i correct that in equality
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00:43:25,840 --> 00:43:30,230
which we proved there holds for any lambda
so i am going to pick one specific lambda
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now
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i am going to pick one specific lambda
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00:43:49,900 --> 00:43:59,990
inner product is a scalar ratio of two scalars
this is y is not zero so since y is not zero
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00:43:59,990 --> 00:44:06,480
this is a positive number and this lambda
is a valid lambda so this should hold for
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00:44:06,480 --> 00:44:34,670
this lambda also for this particular lambda
what is lambda bar
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00:44:34,670 --> 00:44:41,390
is that right i just used the first property
now i am going to substitute this lambda and
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00:44:41,390 --> 00:44:51,810
this lambda bar in the inequality that we
developed earlier so using these lambda and
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00:44:51,810 --> 00:44:59,260
lambda bar i have zero greater than x inner
product x
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00:44:59,260 --> 00:45:08,340
before that let us do a little bit of work
so this implies that minus lambda x inner
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00:45:08,340 --> 00:45:23,130
product y minus lambda bar y inner product
x this is equal to if i just substitute this
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00:45:23,130 --> 00:45:36,230
lambda and lambda bar then what i will get
is that this is nothing but two times x y
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00:45:36,230 --> 00:46:05,690
y x just check what is lambda y x i am substituting
in the first thing there xy and what is lambda
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00:46:05,690 --> 00:46:24,010
bar xy but there it comes yx just algebraic
juggling this is equal to well minus is here
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00:46:24,010 --> 00:46:33,480
of course minus sign will persist so this
is equal to minus two times x inner product
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00:46:33,480 --> 00:46:51,220
y x inner product y bar
y inner product y i will move on to here now
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00:46:51,220 --> 00:47:03,560
is everyone with me on that
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00:47:03,560 --> 00:47:26,460
so this is equal to minus 2 mod
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00:47:26,460 --> 00:47:51,680
now so this quantity here can be now replaced
by our new value so i get 0 greater than x
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00:47:51,680 --> 00:48:17,990
minus 2 and our lambda is
if i substitute for lambda square where lambda
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00:48:17,990 --> 00:48:39,710
square would be
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00:48:39,710 --> 00:48:46,060
if i substitute for lambda square it will
this and then finally the inequality that
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00:48:46,060 --> 00:48:47,740
i get is
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00:48:47,740 --> 00:49:15,500
i finally get an equality which is 0 greater
than x x minus
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00:49:15,500 --> 00:49:22,450
so this minus this is always positive and
i am just doing algebraic juggling now there
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00:49:22,450 --> 00:49:28,320
is nothing specific if you are not followed
right now just goes through meticulously through
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00:49:28,320 --> 00:49:35,700
the notes you will see the steps just substitutions
i have just eliminated by this is a scalar
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00:49:35,700 --> 00:49:43,880
so this square this will cancel with the square
and then you can do the juggling this is not
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00:49:43,880 --> 00:49:46,820
so difficult so what does this imply
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00:49:46,820 --> 00:49:57,050
this implies that the above thing implies
that mod of xy is less than of equal to x
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00:49:57,050 --> 00:50:09,060
x raise to half yy raise to half i take this
on the left hand side and take the square
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00:50:09,060 --> 00:50:19,960
root see this is square of this inner product
of x and y i take this quantity of left hand
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00:50:19,960 --> 00:50:27,850
side because this is greater than this right
otherwise this cannot be greater than 0 and
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00:50:27,850 --> 00:50:32,960
then i am just doing multiplication yy i have
brought it on this side i have just omitted
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00:50:32,960 --> 00:50:38,660
one in between steps everyone clear about
this no problems
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00:50:38,660 --> 00:50:48,220
so what is this this inequality is same as
these results in three dimensions which we
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00:50:48,220 --> 00:50:56,190
know no difference x dot product y mod of
that is always less than than this which is
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00:50:56,190 --> 00:51:05,720
nothing but cos theta less than 1 so i have
proved an equality which is cauchy schwartz
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00:51:05,720 --> 00:51:10,820
inequality i have proved an equality called
cauchy schwartz inequality and this helps
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00:51:10,820 --> 00:51:14,980
us to prove the triangle inequality how will
i prove triangle inequality now
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00:51:14,980 --> 00:51:27,570
what is triangle inequality so triangle inequality
should be norm so we want to prove x plus
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00:51:27,570 --> 00:51:36,560
y to or x plus y less than or equal to norm
x plus norm y we want to prove this inequality
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00:51:36,560 --> 00:51:45,330
finally and i want to use this i want to use
this result this is cauchy schwartz inequality
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00:51:45,330 --> 00:51:54,410
this is generalization of this result well
once i declare x transpose x to be norm of
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00:51:54,410 --> 00:52:02,000
x i can actually even move to this inequality
because this is a scalar i can divide take
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00:52:02,000 --> 00:52:03,000
it inside and so on
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00:52:03,000 --> 00:52:07,660
we will move to that little later in the next
class we will start from this inequality cauchy
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00:52:07,660 --> 00:52:12,369
schwartz inequality and move on to proofing
triangle inequality once we prove triangle
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00:52:12,369 --> 00:52:20,440
inequality we have done once you prove triangle
inequality we have show that inner product
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00:52:20,440 --> 00:52:26,420
defines the norm three axioms of norm two
of them we have already proved the third one
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00:52:26,420 --> 00:52:31,330
was triangle inequality to prove triangle
inequality we need cauchy schwartz inequality
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00:52:31,330 --> 00:52:36,290
but cauchy schwartz inequality not only helps
you to prove triangle inequality it also gives
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00:52:36,290 --> 00:52:44,610
you a way of generalizing definition of angle
it will also give you a way of generalizing
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00:52:44,610 --> 00:52:50,119
so we will be able to define orthogonal vectors
in any inner product space these vectors could
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00:52:50,119 --> 00:52:57,660
be two functions like sin and cos or these
vectors could be two polynomials we will talk
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00:52:57,660 --> 00:53:02,050
about orthogonal polynomials why do we talk
about orthogonal polynomials why do we talk
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00:53:02,050 --> 00:53:03,050
about orthogonal functions
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00:53:03,050 --> 00:53:10,300
they are very very useful when you do mathematics
applied mathematics but why where they called
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00:53:10,300 --> 00:53:15,620
orthogonal why were they called orthonormal
or whatever so those questions will get answered
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00:53:15,620 --> 00:53:21,180
if you understand this basis that is why i
am doing all this proofs so in the next class
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00:53:21,180 --> 00:53:27,670
we will move on to triangle inequality and
then more properties of inner product spaces
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00:53:27,670 --> 00:53:33,210
we will see that the famous pythagoras theorem
which you studied in your eight grade also
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00:53:33,210 --> 00:53:38,560
holds in any of these inner product spaces
what a relief you can work with orthogonal
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00:53:38,560 --> 00:53:39,180
vectors