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Okay Good morning So inequality called as
Caushy Schwaz Inequality
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Now Caushy Schwaz Inequality states that if
I am given in a product space x and I take
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any two elements say x and y that belong to
x then absolute value of inner product between
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x and y is always <= So we proved this fundamental
equality and I said that this was nothing
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but generalization of the result that mod
of cos theta in 3 dimensions we know this
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result this result which we know from 3 dimensions
Caushy Schwaz Inequality was a generalization
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of this particular results from 3 dimensions
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In this particular case x and y are two vectors
that belong to R3 And here so when any two
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vectors x and y that belongs to 3 dimensional
vector space we know this result and this
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particular result is a generalization in any
inner product space okay Now the reason why
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we wanted to work on this particular inequality
was twofold One was well we want to reach
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the concept of angle inner product space in
a general space and at the same time we also
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want to prove triangle inequality in a inner
product space
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Why do I want to prove triangle inequality
I want to define a norm using inner product
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okay So that is why I want to prove the triangle
inequality So how do I prove triangle inequality
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using this particular result So let us move
towards that
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So this result is separate this result is
just for you reference This 3 dimensional
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result is only for your reference So I am
continuing again with our inner product space
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x I am going to take any two vectors x and
y
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So and I want to find out inner product of
x plus y with itself So this would be if I
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just expand this this would be inner product
x plus x inner product y plus y inner product
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x plus y inner product y right Now inner product
a number it could be a positive number or
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negative number inner product they would not
be positive always norm is positive Inner
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product can be positive or negative Cos theta
in this case cos theta can be positive or
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negative
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So I am going to just replace this particular
equality with an inequality So this is <= what
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is x inner product x Always positive not a
problem okay So x inner product x plus 2 times
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absolute of x inner product y plus y inner
product y Do you agree with me Absolute value
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of this this could be a positive or negative
number If this is a negative number absolute
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value is always greater than this value okay
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So I am just replacing this equality with
this inequality fine Even if these are complex
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numbers still this particular inequality will
hold okay
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See now I am going to use Caushy Schwaz Inequality
This absolute value is less than x inner product
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x raise to half into y inner product y raise
to half okay I am going to use this inequality
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here So this will give me x plus y x plus
y inner product is <=
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y inner product y raise to half okay Is that
fine Just using Caushy Schwaz Inequality I
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get this So this is <= actually this quantity
is <= I am continuing This is a square now
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We can see this is a square okay But what
is left hand side So I can write that can
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I say this All are positive numbers this is
a positive number Inner product of a vector
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with itself is a positive number I can take
a square root okay I have expressed the right
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hand side of the square So I can take a square
root What is this This is triangle inequality
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See if I define now
if I define a norm okay which is like this
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then I have all three axioms satisfied What
is the first axiom So we saw this the first
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axiom is norm of x2 is < 0 if x is not = 0
vector and = 0 if and only if x=0 vector okay
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Second we saw what really held was alpha times
x 2 = mod alpha norm x 2 alright And what
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is the third triangle inequality which norm
we have just now proved okay So my third result
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is norm x plus y2 is <= norm x 2 plus norm
y
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That is the result which I have proved just
now right So inner product defines norm very
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nice Inner product this norm is defined using
an inner product or induced by an inner product
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So can we draw now Now that it is a norm it
is a length measure okay Can I extract something
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more out of Caushy Schwaz Inequality So what
was my Caushy Schwaz Inequality Let me see
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whether I can draw some more mileage out of
Caushy Schwaz Inequality
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My Caushy Schwaz Inequality was x y <= right
which is nothing but norm x2 into norm y2
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Everyone with me on this (09:27 minus 09:41
"Voice not clear") This is a positive number
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this multiplication of these two positive
numbers is > this positive number So only
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you can use one way So when you add a higher
number on the right hand side you get inequality
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If you wanted to use minus of something then
probably it is different But there is only
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one way to use it
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When you derive triangle inequality from Caushy
Schwaz Inequality I do not see any other way
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you could use it "Voice not clear" no we are
not assuming I proved Caushy Schwaz Inequality
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I do not know whether you were present in
yesterday's lecture So we proved it no assumptions
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Caushy Schwaz Inequality we have proved by
logical arguments okay And now I am trying
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to see whether I can get some more insights
through it So is this fine
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Now we have defined a two norm okay Now what
is the norm ultimately It is a positive number
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these two are positive numbers right okay
I can take positive numbers inside absolute
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value not an issue right So I won't be wrong
if I say absolute of x fine Is this okay Just
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compare I wanted to have x transpose y divided
by 2 norm of x 2 norm of y take two unit vectors
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in 3 dimensions take inner product what do
you get Cos theta okay
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Take two unit vectors in 3 dimensions and
their inner product will give you cos theta
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Exactly that is what I have arrived at In
any inner product space same inequality no
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difference okay So I am going to say now well
let me define an angle So now let me define
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an angle in any inner product space okay how
do I define an angle
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So let me define two unit vectors x divided
by 2 norm okay Let me define another unit
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vector y cap which is y divided by 2 norm
of y and then cos theta okay Or theta = cos
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inverse of inner product of x is that okay
is this fine So inner product generalization
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of concept of dot product to inner product
allowed me to prove a very very important
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result from 3 dimensions into a any general
inner product space
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So we could define angle between two vectors
Now inner product space could be any set of
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objects like we had set of continuous functions
over an interval okay And when you study your
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undergraduate you come across many such functions
They are called orthogonal functions They
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are called orthogonal polynomials Why are
they called orthogonal What is the basis Okay
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it is basically you are talking because there
is an underlying inner product space
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The inner product defined on it And that inner
product okay allows us to define concept of
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angle between two vectors Vectors as in elements
of the vector space they could be continuous
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functions That is why you know we know all
those results when you look at Fourier series
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See if you take this inner product space set
of continuous functions over minus pi to pi
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okay then we are told when you study Fourier
series that let me define the inner product
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here inner product is defined between any
F and G two functions as integral minus pi
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to pi ftgtdt
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Let me take this inner product space okay
and let me define this particular product
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Well what we are told is that inner product
of sin t cos t is 0 because they are orthogonal
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okay When you hit up on this consult first
time that two functions why are they orthogonal
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in what sense Because when you think of orthogonality
you are trained to think in terms of 3 dimensions
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i j k and so on okay But what you should realize
is that is in the same sense i j k three unit
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vectors in 3 dimensions are orthogonal
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These two vectors are orthogonal in that in
a product space set of continuous functions
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over minus pi to pi or if you change 0 to
2 pi So this is interesting So this allows
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us to talk about orthogonality of functions
Orthogonality of general vectors in any vector
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space in any inner product space So far so
good So we have defined angle obvious thing
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that comes next is orthogonality okay
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We say that two vectors are orthogonal
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inner product is 0 simple If the inner product
is 0 then these two vectors x and y any arbitrary
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vectors x and y for which inner product is
0 in a product space they are orthogonal vectors
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okay What are the other concepts that you
need when you start orthogonality Well one
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thing we talk about is that a vector is perpendicular
to a plane right We have to use an ocean of
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vector being perpendicular to a plane or a
set
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Let us say so this if I have this plane okay
I can say that this plane okay this particular
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vector is perpendicular to every vector in
this plane right every vector in this set
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I could talk about a entire plane I could
talk about this limited set and this particular
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vector will be perpendicular to all the vectors
in this set okay So if you have a subset as
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which is subset of inner product space s and
a vector x that belongs to inner product space
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is such that x is perpendicular to y for any
y that belongs to s then we say that x vector
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is perpendicular to s
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If s is some subset of inner product space
okay And I take any arbitrary vector x in
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the original space This is a subset this is
a vector if this vector is perpendicular to
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every vector that belong to the set then the
vector is perpendicular to the entire set
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All these concept will require more and more
once we progress okay So what result that
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I wanted to generalize what was the best result
in geometry that you keep using all the time
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Pythagoras theorem okay
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Can you prove Pythagoras theorem What is Pythagoras
theorem What is the statement of Pythagoras
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theorem In 3 diemnsions let us look at 3 dimensions
what is the statement How will you state Pythagoras
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theorem in 3 dimensions If you are given any
two vectors x and y okay
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I am given any two vectors x and y in 3 dimensions
and let x be perpendicular to y How will you
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state Pythagoras theorem (18:33 minus 18:44
"Voice not clear") x and y are two vectors
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what will form an triangle x plus y will form
the triangle x minus y also can form a triangle
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both will hold Then so you want to say that
can I say this is = x transpose x plus y transpose
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y as he is rightly pointing out this could
be said even for x minus y okay
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So x minus y transpose x minus y will also
give you x transpose x plus y transpose y
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So this is my Pythagoras theorem in 3 dimensions
Do you agree with me has anyone has doubt
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here What is x plus y if I take two vectors
in 3 dimensions Just try to visualize unless
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you visualize you will not get it
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Let us say this is x vector and this is y
vector What is x plus y Parallelogram law
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in fact now I would expect the Parallelogram
law to hold in here Do you remember Parallelogram
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law Everything will hold I mean you are just
generalizing This is my x plus y vector right
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and this is x minus y vector okay So now if
x and y are perpendicular okay then what we
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are saying is square of so we are looking
at a
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scenario where x and y are like this and x
plus y is actually this right
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So square of length here is this square plus
this square that is all I am stating here
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okay Is this fine So now all that I need to
generalize this in inner product space is
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to use the concept that if two vectors in
inner product space are orthogonal then their
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inner product is 0 I just start with the same
thing so in an inner product space x let x
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and y belonging to x well I am writing everything
in this cryptic language because it is faster
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to write otherwise and you will get used to
it after some time
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X is perpendicular to y right I pickup two
element x and y in a inner product space which
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are perpendicular All that I have to do to
prove Pythagoras theorem is to start with
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x plus y this is = norm x this is = x inner
product x plus x inner product y plus y inner
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product x plus y This is 0 this is 0 x and
y are perpendicular inner product is 0 okay
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What follows is the classic Pythagoras theorem
generalized to any inner product space a grand
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generalization of ideas
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Same ideas what you should not forget is the
ideas of geometry which we are using from
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your school 3 dimensional vector space which
you are used to in your college same thing
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is being generalized in different spaces okay
So if your geometrical ideas in 3 dimensions
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are clear you will understand what is happening
here You cannot visualize what exactly this
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means in a function space visualization is
not so easy
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I don’t know whether it is possible or in
n dimensions or but geometrically it is the
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same thing what is happening here when you
have 2 perpendicular vectors and writing orthogonal
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Pythagoras theorem geometrically it is not
at all different That is important to understand
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okay same geometrical concepts are this We
are not able to visualize this okay So what
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next how did orthogonal vectors help you in
3 dimensions what were they used for
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Can somebody show light Standard basis orthogonal
basis very very useful right We use orthogonal
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basis all the time So you had three unit vectors
which are orthogonal in fact you chose them
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ortho normal What is ortho normal Unit vectors
Ortho normal their magnitude was 1 okay So
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ortho normal vectors helps us to define any
arbitrary vectors in terms of its components
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along certain directions right
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So we write a vector x component along I direction
and y component along j direction This is
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the first time when you start looking at coordinate
geometry This is how you start representing
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a vector right So we need now exactly the
same thing we need to generalize a set of
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ortho normal basis vectors in any inner product
space and then we should be able to express
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a given vector in terms of a orthogonal basis
right Because orthogonal basis has many many
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advantages in computations as compared to
non minus orthogonal basis okay
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See in 3 dimensions how many ways you can
construct a basis What is a basis in 3 dimensions
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So for example 3 independent vectors okay
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So in 3 dimensions just like this let us say
k this is unit vector here say k i and j just
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like these 3 unit vectors form a basis okay
I can take some 3 other vectors say e1 e2
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e3 as long as these 3 vectors are linearly
independent they can form a basis There are
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infinite ways of defining a basis in 3 dimensions
Given that there are infinite ways of defining
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if I give you some arbitrary vectors in 3
dimensions say this vector v okay I can write
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vector v as v = day x1e1 plus x2e2 plus x3e3
where e1 e2 e3 are 3 basis vectors
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I am perfectly allowed to do this okay Yet
we prefer to work with so same vector v we
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find it convenient to write in terms of some
component xi yj plus zk and so on So we prefer
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this basis over this basis We have this basis
over this basis okay So likewise are there
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some special basis which are more useful when
you do computations it turns out that there
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are okay Now for example I have a continuous
function which is the polynomial okay
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Can I define a orthogonal basis for a set
of polynomials then I can express a polynomial
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something like this using orthogonal components
I can express a function using orthogonal
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components along certain orthogonal polynomial
directions So I am going to generalize this
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idea I don’t like this when I work in 3
dimensions I prefer this So I need orthogonal
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basis when I go to an inner product space
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So now I am generalizing a concept of orthogonal
set in this case i j and k is a set of orthogonal
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vectors In fact they are set of ortho normal
vectors right So in inner product space if
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I give you a set okay and if I pick any two
elements in that set okay and if the inner
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product of any two elements is 0 then that
set is orthogonal set When we will call it
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as ortho normal set If each one of them is
a unit vector then it is a ortho normal set
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So if each vector has unit magnitude then
we call this set as a ortho normal set okay
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Now do you remember how do you construct if
I give you in 3 dimensions if I give you 3
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vectors which are not orthogonal I want to
construct an orthogonal set starting from
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a non minus orthogonal set how do you do it
Does if I say Gram minus Schmidt Process does
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it ring a bell No you don’t know what is
Gram minus Schmidt Process
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Okay we will study what is Gram minus Schmidt
Process So I like orthogonality because it
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helps me to represent vectors in a very way
And so if I am given a set which is not orthogonal
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I would like to construct a set which is orthogonal
okay If I am given a set which is not orthogonal
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then I can construct an orthogonal basis which
helps me two represent vectors okay
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First I am going to start looking at 3 dimensions
we will generalize to setup polynomials we
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will then go to functions functions space
and on So okay let us go back to our 3 dimensional
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vector space
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We have this vector v here and so let us call
this x y and z directions and this our i j
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k are three unit vectors okay If I wanted
to compute component of v along x how do I
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do it Dot product with unit vector in that
direction right So
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I use the unit vector we transpose i So this
vector this will give me inner product dot
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product actually we should not say v transpose
v i that is the right So dot product will
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give me x component okay
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Dot product of v with j will give me y component
and dot product of okay So suppose I find
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out if I am given vector v okay I find out
the component along x okay Let's call this
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vector as vx what is vx Vx is component along
x I am going to call it as vx okay So what
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will be v minus vx What will be this vector
v minus vx It will be two components that
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are remaining along y and z directions So
everything that was along x has been removed
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okay Now what remains is only
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So in fact you would expect that component
to lie in which plane Yz plane right okay
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Now this idea I am going to use to come up
with this concept of Gram minus Schmidt Process
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okay Is this clear what I talked about just
now that you find a component along a particular
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direction remove it from the original vector
what remains is along the remaining orthogonal
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components okay So this is the very very important
concept
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Gram minus Schmidt orthogonalization can be
done only in an inner product space not in
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any vector space because inner product defines
angle orthogonality and the things that you
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really need to construct an orthogonal basis
okay Idea of orthogonal basis cannot be thought
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of in some other arbitrary vector space where
inner product is not defined okay So definition
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of inner product is crucial when it comes
to okay Now let us start with R3
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So I am taking three vectors v1 v2 v3 which
are linearly independent in R3 but not orthogonal
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okay they are not orthogonal they are just
okay I am given three vectors in R3 and then
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I want to construct a set which is orthogonal
basis right I could have constructed a basis
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from this which is a non orthogonal basis
This basis would be one way of constructing
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a non orthogonal basis will be v1 up on norm
v1 right and v2 up on norm v2 and v3 up on
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norm v3
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I can construct a unit vector I can construct
three unit vectors but they are not orthogonal
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okay So I would like to go to orthogonal set
from this okay
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So let us define vector e1 okay This vector
e1 I am going to define as v1 divided by 2
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norm of v1 Is this fine Okay Now I want to
construct this is unit vector So I got one
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unit vector I want three unit vectors which
are orthogonal In fact I would like them to
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be ortho normal Then what I am going to do
is I am going to remove the component I am
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going to define a new vector z2 okay which
is v2 minus component of v2 along e1
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How do I find component of v2 along e1 Dot
product times this is a scalar right this
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is the component this is the scalar component
along e1 So this vector minus this will be
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everything now that is left which is not along
So z2 will have everything that is not along
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e1 Is e1 perpendicular to z2 You can just
check that z2 dot product e1 what is this
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This is v2 dot product e1 minus v2e1 dot product
e1 or e1 inner product e1
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What is e1 inner product e1 1 So this is 1
So what do you get here 0 okay So I have constructed
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a vector z2 which is orthogonal to okay
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So z2 is perpendicular to e1 But z2 is the
vector which is not a unit vector I like unit
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vectors i j k okay So how do I get a unit
vector I will take e2 which is z2 divided
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by norm z2 okay So I have two vectors See
this z2 and e2 are aligned along the same
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direction magnitudes are different right So
e2 and e1 are also perpendicular So e1 is
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perpendicular to e2 okay Is that fine Now
what next I want to now construct a third
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vector so v3
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So I will construct a vector z3 which is v3
minus component along e1 You can very easily
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check that Z3 is perpendicular to e1 z3 is
perpendicular to e2 Not difficult to check
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okay Just take inner products we will see
that z3 is perpendicular to e1 z3 is perpendicular
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to e2 okay So e1 e2 e3 are mutually orthogonal
okay So how do I create e3 now Take a unit
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vector along z3 okay
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So we started with a non minus orthogonal
set and we got an orthogonal set I can do
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this why just in 3 dimensions You have some
doubt Which one See this e3 is a third vector
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which I am going to define just by taking
unit direction along z3 okay See I started
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with What did I start with I start with v1
v2 v3 these are not orthogonal okay So from
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v1 I constructed this e1 vector okay Then
I removed component along e1 from v2 okay
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Whatever was left was perpendicular to e1
okay
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Next then I defined this z2 I defined a unit
vector along z2 okay Then I removed component
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of v3 along e1 v3 along e2 right Whatever
was left was perpendicular to both e1 and
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e2 We can just check this See because e1 and
e2 are orthogonal if you take inner product
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of e1 with e2 will be 0 And inner product
of e2 with e2 will be 1 okay So it will just
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nicely follow So you started with 3 non orthogonal
vectors finally I got this z3 which is not
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a unit magnitude vector
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So I am just making this a unit magnitude
vector here okay So I can generalize this
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process in n dimensions if you are given n
vectors in n dimensions okay how could I construct
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an orthogonal set
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In n dimensional space how can I go on doing
this Gram minus Schmidt orthogonalization
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So I could systematically go from 1 2 3 4
and so on So this is e1 is x1 then z2=x2 minus
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inner product x2e1e1 and e2= and so on I just
go on methodically doing this same thing okay
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Then e3 then e4 then e5 then e6 I can go up
to en So starting from a non minus orthogonal
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set okay so what we will see in the next class
is we will take an example in 3 dimensions
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construct an orthogonal set
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We will take a set of polynomials which are
not orthogonal construct set of orthogonal
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polynomials If you just follow Gram minus
Schmidt Process what will pop out is legendre
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polynomials okay I think you have heard this
name legendre polynomials And then you must
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have heard shifted legendre polynomials And
then you must have heard Bessels polynomials
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All these things will fall into line if you
understand Gram minus Schmidt Process okay
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It is some orthogonal set constructed on some
inner product space of interest okay and those
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sets can be constructed simply following the
simple idea from 3 dimensions Gram minus Schmidt
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orthogonalization okay That is the message
okay So next class we will look at examples
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of this