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under this broad title i am going to look
at 5 0r 3 or 4 sub areas first of all we will
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define probably what is the vector space then
we move to nonlinear spaces and also what
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are called as banach spaces then we have now
inner product spaces and we will also catch
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up on hilbert spaces so these are names of
these among famous mathematicians and then
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we look at gram schmidt process in the end
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so these are the 4 sub areas which i need
to cover now gram schmidt spaces as applied
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to any general vector space now what is the
vector space what did we know about vector
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spaces when we start thinking about vector
spaces from what background we have background
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at we have from undergraduate we either look
at well we normally imagine a vector in a
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3 dimensional space
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so this is the vector say x or let us call
this vector v let us 3 components x y and
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z and this is how we longer we imagine a vector
space now what is done in functional analysis
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you should distilled essential properties
of this vector space and then come with a
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new definition called vector space which is
more generic which can be applied to any set
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of objects any set of objects that are irrelevant
to us when we do computational or analytical
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mathematics
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now actually the other course that we are
doing in that also in the beginning there
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will be some introduction into these vector
spaces so now what is it that we need to generalise
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and why we need do generalise first of all
we can look at this vector space or set of
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all vectors in this vector space so if i had
said okay as i said on which certain operations
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can be done okay what are these operations
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addition you can have 2 vectors and get a
vector and a nice thing is that you get a
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vector in the same space and you can multiply
a vector by a scalar and you get another vector
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in the same space so these are e generic properties
of any two vectors in the space and i could
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use this to generalize define a generalized
notion of a vector space it is not enough
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to just generalize this notion of a vector
space you also need something more to work
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with vector spaces
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we need to know about the length of a vector
because that is a critical thing that we use
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when we actually work with vectors in 3 dimensions
so we need to have a generalization of that
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which is called as norm of a vector we will
talk about norms of vector it is not enough
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to just talk about norm we have notions of
a sequence in one dimension so a sequence
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which is converging to a something called
limit
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so what is the limit in n dimensions or 3
in dimensions so we need to actually generalize
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the concept of conversions and limit that
there are some funny things that happen when
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you start working with 5 dimensional vector
spaces and that is where we had i had mentioned
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is banach spaces and hilbert spaces so these
are some special category of spaces which
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we are going to look at
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now it is not enough just to work with norm
and conversions we need something more what
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else we use in 3 dimensions what is the important
geometry concept that you need professor student
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conversation starts not coordinate coordinate
is of course that was define the space many
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times a coordinate system will come it is
not coordinates its angle angle between 2
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vectors so very very important concept professor
student conversation ends
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well in 2 dimensions everything comes together
package we do not really think of these things
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separately but when you generalize this concept
to any other space we need to make efforts
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to define what is angle between 2 vectors
and also one of the most important concept
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that we used in 3 dimensions is orthogonal
vectors 90 degrees 2 vectors and then well
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of course the pythagoras theorem which is
used in many many ways
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so what we are going to look at initially
is it possible to generalize this concepts
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and develop some notions of vector spaces
on generic sets which are useful in mathematical
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analysis why am i doing all this in beginning
of the course which is supposed to be computational
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methods and you would be starting with there
was a bit of recipes well if you understand
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these grand generalizations which were probably
done in the beginning of 19th century beginning
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of 20th century
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then it becomes very easy to understand the
foundations of different memorial methods
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that we are going to study so this 6 or 7
lectures which might look disconnected in
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the beginning are actually deeply connected
with what we are going to do later okay so
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this forms the foundation and you will understand
basis of many many methods if you understand
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this concept of vector spaces orthogonality
and so on
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many of these things are unknowingly used
when you do undergraduate courses without
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you know being given a thorough explanation
here will lay a systematic foundation of vector
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spaces now let us well should i want to talk
about you know 4 dimension or 5 dimension
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10 dimensional spaces and then i will also
move to something called infinite dimensional
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spaces and well it is not possible to visualize
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in fact it is not possible to visualize anything
beyond 3 dimensions you cannot visualize the
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4 dimensions or 5 dimension spaces and obviously
not an infinite dimension spaces so these
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are word of caution before we move into this
is that it is enough to know your geometry
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school geometry well it is enough to know
your undergraduate 3 dimensional word well
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if you understand the concepts in the undergraduate
3 dimensional word or your school geometry
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it is you will understand everything i am
doing okay
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it is only a matter of generalizing these
concepts same concept which have been use
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in right since our eighth standard where one
generalise into something very very elegant
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so now let us begin with let us begin with
the concept of closure a closure of a particular
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set for an operation is defined as if you
take any two elements from the set let x be
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a set let x is a set and there are any 2 elements
say x and y belong to x okay
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then x operation y also belongs to the same
x now this operation here i have written as
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plus it could be any operation it could be
multiplication it could be division okay so
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given any set if you take any two elements
of this set and if you perform an operation
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for example multiplication okay and if the
element that results after performing the
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operation also belongs to the set then it
is called a closed set okay
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for example set of integers is closed on addition
let us closed set will be division right the
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next concept that is important is a field
what is the field professor student conversation
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starts and division correct so field is the
set of elements close under
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so well the well known examples of field set
of real numbers set of complex numbers and
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these are the two fields which have been they
are going to use
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so they are going to denote these as r r we
denote as the set of real numbers and c let
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we put the set of complex sums that is closed
under 2 operations now a vector space is a
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00:11:05,839 --> 00:11:21,829
set of objects it is closed under addition
so if i take any element x and y belonging
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to x then x plus y also belongs to x but it
is not enough to have this a set of objects
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we need something more to define a vector
space we also need a field
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for example a field could be r so let us we
call the field as f so we need two things
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here a set of objects x and a field f if i
take any scalar alpha from f and any x belonging
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to x then alpha times x is called a scalar
multiplication this also belongs to x so our
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vector space a conservative generalization
of 3 dimensional vector space is nothing but
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nothing but a set of objects which actually
satisfy these 2 operations
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or a set which is closer under these two equations
given the field f okay it depends upon the
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combination so these x and f they are a combination
you can never separate them you have to consider
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them together well let me start generalizing
and giving you examples of spaces which are
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my first example is going to be x corresponds
to rn and f corresponds to r what is rn rn
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is n double okay or a vector which has n components
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so my general vector here x that belongs to
x will be represented as it has n components
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x1 x2 xn okay it may have n components can
you give me an example there you need a such
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a thing let us say i am dealing with a i am
dealing with some chemical reactor and i decide
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to associate a vector space that defines different
variables so here x1 x2 to xn could consist
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all for example the xn could be temperature
in the reactor x2 could be pressure and x3
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to xn could be different chemical species
that are present inside the reactor
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00:14:00,889 --> 00:14:11,741
so this is a vector of that represent the
state inside the reactor or let me a take
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a distillation problem let us say distillation
problem will have different trace and on each
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tray you have a temperature pressure composition
right so if there are 20 trays and it is binary
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distillation problem how many variables you
will expect to have 20 temperatures 20 pressures
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actually pressure will be varying across the
column when compositions about 60 well there
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00:14:40,629 --> 00:14:42,790
are correlations between y and x
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so there would be 80 elements in a vector
or 60 elements in the vector that defines
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all the variables and are associated with
an distillation problem which has 20 trays
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final distillation problem which also i can
think of examples from chemical engineering
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which would actually have where you deal with
vectors at a higher dimensional vectors where
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an example that will tell you what is which
combination will not be a vector space
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see for example if i take x to be rn and f
to be c set of complex numbers professor student
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conversation starts will this form a vector
space no why professor student conversation
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ends the scalar multiplication will break
down if i take a scalar which is complex multiplied
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to a vector which is the real value i learned
it element from x so this is not a vector
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space when my next example is my next example
is little unconventional
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so now i am going to move to set of real value
matrices so this is my x this corresponds
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to my x okay and my f is set of real numbers
set of real value matrices professor student
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conversation starts let us this form a matrices
why professor student conversation ends if
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you take any two n cross matrices if you add
them you still get an n cross n matrix if
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you take a scalar and multiply it to an n
cross n matrix you still get an n cross n
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matrix okay
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so if i take any two matrices say a and b
which belong to f then i can say that any
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alpha times a plus beta times b belongs to
also x sorry this is x here and alpha beta
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belong to f so for any matrix x for any matrix
a and any matrix b which are both n cross
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n so these are elements these are vectors
in this space vector space if i take a scalar
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multiplication of a and a scalar multiplication
of b then this sum should also belong to this
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space which is true for any n cross n vector
okay
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so and my fourth example is something that
i am going to use quite often in this course
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so my example number 4 is this is denoted
as c a b so c a b is set of continuous functions
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set of real valued continuous functions set
of all the real valued continuous functions
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on interval a b so if i have a function say
ft which and a function gt both of which belong
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to x then and if i take any if i take any
two scalars say alpha and beta that belong
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to f
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then alpha times ft plus beta times g of t
also belongs to x set of real value have you
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come across this kind of functions where where
did you study this assumption fourier series
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not ever what happens in fourier series you
talk about you talk about functions on minus
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pi to pi or 0 to 2pi remember something rings
a bell and b are 2 constants have you going
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00:19:07,560 --> 00:19:16,040
to look for fourier series much more detail
in next 2 lectures okay
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00:19:16,040 --> 00:19:23,540
so you are agreed with me if i take a continuous
function and multiplied by a scalar it is
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simply a continuous function let us see the
continuous function right if i add 2 continuous
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functions will the addition be continuous
if f of t is continuous and g of t is continuous
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add 2 continuous functions i still get a continuous
function so scalar multiplication vector addition
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both properties hold in this abstract set
it is the difficult to visualize
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how this set looks like you know we are used
to visualize it in 3 dimensions nevertheless
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the property is that hold the fundamental
properties that hold in 3 dimension also hold
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in the same that is very very important okay
so if you have a vector if you have a vector
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with which holds this side multiply it by
a scalar you get a vector in the same side
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it is very important and g of vector which
is well this is not enough to just define
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vector spaces
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we also have to talk about subspace so very
very important concept in 3 dimensions what
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is the subspace in 3 dimensions what are the
sub spaces that we go in the dimensions professor
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student conversation starts line passing through
a yeah correct it is only subset a plane so
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whether the plane not passing through a will
it be a sub space right why good professor
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student conversation ends
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let us define what is
a subspace if i want to define the subspace
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so this is i have a set of vector x and field
f and then let m be the subset of x and with
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some subset of x and nonempty subset of x
and if i take any 2 elements in m so x and
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y they belong to m and alpha beta belong to
f then alpha x plus beta y should belong to
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m and the way we define subspace is a nonempty
subset a nonempty subset of original space
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x okay like example that he gave us now
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a line passing through origin or a plane passing
through origin okay but why was origin important
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that the origin is required in this space
is hidden in this definition can you denote
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any word is that you know the main thing is
any alpha beta belonging to f so what about
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0 0 if i choose also alpha to be equal to
0 and beta to be equal to 0 then 0 times any
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vector plus 0 times any vector give me 0 vector
0 vector should be contained in the space
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okay
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so if i have a subspace then it follows from
this definition that the 0 vector the origin
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should be contained in the space so only those
sets only those sets which contain the origin
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qualified to be subspaces okay now let us
understand this little more if i take okay
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let me try to draw a subspace imagine that
this square the plane which passes through
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this this is a plane that passes through the
origin okay now that 2 situations 2 scenarios
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one is this is a finite this is a finite set
it is only like a piece of paper okay let
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us look at this piece of paper which is passing
through the origin it is finite size it is
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passing through the origin will it form a
subspace professor student conversation starts
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just because it passes through the origin
will it form a sub space i can take 2 elements
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this is passing through the origin i can take
2 elements such that x plus y not for all
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x plus y correct
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the word all is important it should happen
for every x plus y okay it may happen that
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if i take a finite set like this and not the
infinite set it may happen that for 2 vectors
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say this vector and this vector the addition
may not belong to this small set yeah but
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vector spaces origin is included it is obvious
the subspace is a small set subspaces is a
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smaller set does every smaller set qualified
to be a subspace is the question i will answer
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so what is for example what is zero elements
in this what is zero here yeah constant function
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ft equal to 0 over interval a to b okay here
it is important to remember that t belongs
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to a b component well do we get this kind
of functions think of temperature profile
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in the heat exchanger okay my a would be 0
to 1 where it will not be time t do not associate
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t with time it would be space so z is my special
variable okay varies from 0 to t okay
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and i can look at the temperature profile
inside but distillation inside the heat exchanger
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is it a continuous option yes it is a continuous
option so this kind of vectors these kind
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of vectors spaces are very very commonly encountered
in chemical engineering examples and of course
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the zero the zero element would be well nonzero
temperature we often talk about perturbations
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and in steady state
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if you have a steady state and a perturbation
the perturbation vector is zero which means
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you have a steady state so you may have you
will not have in the case of heat exchanger
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example the zero element technically would
be everywhere you have a zero temperature
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such as does that exist but the space in the
space you can of course define zero vector
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which is so coming back to this subspaces
every subset does not qualify to be a subspace
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okay
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00:26:41,770 --> 00:26:48,790
this thing is important if i take alpha any
scalar alpha multiplied by vector then the
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00:26:48,790 --> 00:26:52,601
resulting vector also should be included inside
the space so if i take this vector and if
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i multiply it by a large scalar the new vector
will be here which is not included in this
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small finite set this is not a subspace so
just going through a origin is not sufficient
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00:27:02,210 --> 00:27:09,830
okay you need to have closure we need to have
closure of this operation alpha times x
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00:27:09,830 --> 00:27:14,770
for any alpha beta times y this sum should
be belong to so can i give you an example
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00:27:14,770 --> 00:27:17,100
i will just give you an example which is completely
different but generalizes this concept let
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us look at this space okay set of polynomials
set of polynomials are they continuous functions
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00:27:22,210 --> 00:27:30,050
yes right set of polynomials is a continuous
function okay set of polynomials let us say
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00:27:30,050 --> 00:27:39,250
define over a to b or 0 to 1 if you want to
fix imagine 0 to 1
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00:27:39,250 --> 00:27:48,650
so set of polynomials defined over 0 to 1
what will be the set so let me take this set
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00:27:48,650 --> 00:28:02,570
let me take this set s we call it m here let
us be this side be m which is 1 t t squared
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00:28:02,570 --> 00:28:15,450
or t to the power n okay will this combination
be also polynomial this is an nth order polynomial
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00:28:15,450 --> 00:28:32,130
okay this is an nth order polynomial now when
i am visualizing each one of them as nth order
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00:28:32,130 --> 00:28:37,680
polynomial with some coefficient 0 okay
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00:28:37,680 --> 00:28:50,730
so this set of all possible polynomials with
any alpha 1 to alpha n set of field this particular
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00:28:50,730 --> 00:28:59,080
set will form a subspace of this vector space
because these are continuous functions these
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00:28:59,080 --> 00:29:03,980
are continuous function and then if you take
a polynomial if you take a polynomial finite
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00:29:03,980 --> 00:29:10,590
order polynomial add to that polynomial and
we will get a finite order polynomial okay
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00:29:10,590 --> 00:29:15,520
so all those properties all these properties
with hold for set of polynomials and then
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00:29:15,520 --> 00:29:22,130
you can show that this is subspace 0 element
will be there zero function zero polynomial
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00:29:22,130 --> 00:29:23,130
okay
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00:29:23,130 --> 00:29:28,210
so all the things that you need will be there
in this thing okay and i will another example
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00:29:28,210 --> 00:29:39,809
of subspace can you think of a subspace for
n cross n matrices for example let us take
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my set here so you understand this if you
understand these examples because just writing
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the definition is too abstract unless you
associate with some real examples it is not
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00:29:51,430 --> 00:29:56,170
possible to understand these concepts
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00:29:56,170 --> 00:30:14,530
so like by x be set of and of course my field
is r okay i am going to define a subset m
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which is a subset of x which is a non empty
subset of x so m here set of all symmetric
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00:30:26,590 --> 00:30:36,710
matrices is it a subspace what is a sub space
if i take a scalar alpha see when you call
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00:30:36,710 --> 00:30:45,110
the matrix to be symmetric a transverse is
equal to a okay let us test this you know
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00:30:45,110 --> 00:30:53,800
alpha times a transpose will it be equal to
alpha times a for any alpha right
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00:30:53,800 --> 00:31:04,820
for any alpha by the way what is the zero
element in this space what is the zero vector
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00:31:04,820 --> 00:31:13,860
dull magnets so if i set alpha to be 0 it
is a symmetric matrix right it belongs to
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00:31:13,860 --> 00:31:26,070
a subspace what about alpha if i take any
of 2 matrices a and b belonging to this subset
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00:31:26,070 --> 00:31:34,290
m okay with their linear combination also
belong to m if this is symmetric and if this
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00:31:34,290 --> 00:31:41,020
b is symmetric will this addition also be
symmetric it is a symmetric matrix
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00:31:41,020 --> 00:31:53,050
so this is subspace on the subset defined
by this defined by this either you know it
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00:31:53,050 --> 00:32:01,070
is a set which forms the sub space not every
set will form a sub space but particular set
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00:32:01,070 --> 00:32:08,470
of set of symmetric matrices will form likewise
if you go to a set of complex valued matrices
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00:32:08,470 --> 00:32:15,880
with field to be complex numbers you can define
with hermitian matrices set of all hermitian
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00:32:15,880 --> 00:32:25,610
matrices okay will be a subspace of the set
of complex valued matrices okay
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00:32:25,610 --> 00:32:30,500
so these are the generic example what is the
next thing that will be when you start looking
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00:32:30,500 --> 00:32:39,160
at vector spaces what is the thing that i
use most well one of the most important concepts
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00:32:39,160 --> 00:32:48,700
that we use is basis and dimension what is
the dimension of vector space what is dimension
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00:32:48,700 --> 00:33:01,010
how do we define dimension this is the common
number of coordinates okay
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00:33:01,010 --> 00:33:20,920
let us look at this subspace this is my this
is a line passing through the origin and all
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00:33:20,920 --> 00:33:26,340
of us agree that this is a subspace okay so
line passing through the origin any vector
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00:33:26,340 --> 00:33:31,400
of this line will be represented by 3 components
does it mean it is 3 dimensional subspace
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00:33:31,400 --> 00:33:46,490
what is the dimension why number of independent
vectors are only 1 so just because a vector
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00:33:46,490 --> 00:33:54,470
has n components does not mean that you know
the dimension of the vector of the vector
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00:33:54,470 --> 00:33:55,770
space or a subspace professor student conversation
ends
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00:33:55,770 --> 00:34:02,850
the dimension of this subspace is only 1 okay
and there is only 1 independent direction
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00:34:02,850 --> 00:34:10,220
let us say you call this some vector x so
any vector let us call this vector b so any
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00:34:10,220 --> 00:34:16,879
vector on this line will be alpha times v
okay if alpha is minus it will go in this
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00:34:16,879 --> 00:34:21,559
direction so alpha is going in that direction
so but basically it is alpha times it is only
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00:34:21,559 --> 00:34:27,249
one independent direction okay so likewise
is a plane cutting or a plane passing through
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00:34:27,249 --> 00:34:34,539
the origin what is the dimension of the space
2 because there are only 2 independent vectors
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00:34:34,539 --> 00:34:43,450
two independent vectors can generate the entire
space so we need to now generalize this concept
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00:34:43,450 --> 00:35:01,019
of dimension well to generalize this concept
we need we analyse many other concept we have
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00:35:01,019 --> 00:35:14,990
to have notion of linear combination defined
after that we will have to define what is
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00:35:14,990 --> 00:35:22,970
called as a basis set and then move on to
so you should go through the notes i have
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00:35:22,970 --> 00:35:25,609
given many more examples of vector spaces
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00:35:25,609 --> 00:35:30,309
so and now as i said i am not going to write
everything onto the board we should look at
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00:35:30,309 --> 00:35:39,099
the notes now if i am given the vector now
i have to introduce one important notation
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00:35:39,099 --> 00:35:46,799
here because they are going to work with set
of vectors and each vector might be n double
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00:35:46,799 --> 00:35:57,160
okay each vector might be n double so i have
to introduce a new notation now i am going
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00:35:57,160 --> 00:36:04,300
to consider a set here a set xi
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00:36:04,300 --> 00:36:13,099
i have some space x here and this xi these
are vectors that belong to this set x where
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00:36:13,099 --> 00:36:32,180
i was from 1 to n okay it is quite possible
that my set is nothing but rm so m doubles
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00:36:32,180 --> 00:36:39,010
okay so an element here there are n vectors
and each one of them is an m double okay so
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00:36:39,010 --> 00:36:59,930
where i am going to define this is xi it corresponds
to xi1 xi2 xin so this notation is central
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00:36:59,930 --> 00:37:06,749
to our course i am going to use it very very
often then you have a sector
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00:37:06,749 --> 00:37:17,239
so superscript in brackets is used to indicate
ith vector okay and the subscript is used
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00:37:17,239 --> 00:37:25,660
to indicate the component of the ith vector
okay so xi2 is second component of ith vector
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00:37:25,660 --> 00:37:36,329
and so on so we will be using this notation
very very often now if i choose any set of
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00:37:36,329 --> 00:37:48,499
scalars if i choose any set of scalars say
alpha1 alpha2 alpha n then a vector which
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00:37:48,499 --> 00:38:06,150
is defined by alpha1 x1 plus alpha 2 x2 this
vector is called as linear combination of
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00:38:06,150 --> 00:38:07,150
this set of vectors
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00:38:07,150 --> 00:38:18,869
this is the set of vectors belonging to
space x okay alpha 1 to alpha n or some scalars
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00:38:18,869 --> 00:38:26,200
arbitrary scalars belonging to the field f
and then the vector that you get by alpha
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00:38:26,200 --> 00:38:36,059
1 times x1 plus alpha 2 times x2 up to alpha
n times xn this particular vector obviously
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00:38:36,059 --> 00:38:42,540
we are dealing with a vector space so or we
are dealing with a subset in which this linear
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00:38:42,540 --> 00:38:51,509
combination we should talk about a subset
alone then it is a finite set
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00:38:51,509 --> 00:38:58,279
and if we take all possible linear combinations
of these vectors they give a special subset
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00:38:58,279 --> 00:39:18,569
that is called a span professor student conversation
starts this is a vector this is ith vector
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00:39:18,569 --> 00:39:26,700
okay see for example i may have 2 vectors
let me take a 5 dimensional space so i have
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00:39:26,700 --> 00:39:45,289
vector one okay 1 2 3 4 5 and vector 2 which
is 5 4 3 2 1 okay now how do i refer to third
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00:39:45,289 --> 00:39:46,369
element of vector 2
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00:39:46,369 --> 00:40:02,700
so i will say v2 3 that is equal to okay vector
2 third element okay similarly v2 5 will be
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00:40:02,700 --> 00:40:11,829
no no this is just a notation just a notation
that which we are going to very very often
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00:40:11,829 --> 00:40:18,730
whenever you have subscript in brackets or
superscript in brackets it implies that ith
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00:40:18,730 --> 00:40:26,099
vector okay and if i want to refer to this
zth component of a ith vector okay then i
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00:40:26,099 --> 00:40:32,250
will use xi z it is not a matrix it is a rotation
professor student conversation ends
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00:40:32,250 --> 00:40:37,299
now this kind of things do appear in numerical
methods in computations because we do introduce
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00:40:37,299 --> 00:40:41,759
procedures okay we start from one vector and
then you get another vector and another vector
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00:40:41,759 --> 00:40:46,489
okay so you have a sequence of vectors and
that is where you need to know this have this
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00:40:46,489 --> 00:40:53,440
is a little complex notation so sometimes
we develop algorithms in which we need to
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00:40:53,440 --> 00:41:00,140
worry about this superscript and in this subscript
together okay that is why we need to have
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00:41:00,140 --> 00:41:01,729
this notation okay
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00:41:01,729 --> 00:41:08,279
so this span is set of all possible combinations
understand set of all possible combinations
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00:41:08,279 --> 00:41:24,780
if i give you 2 vectors in three dimensions
if i give you 2 vectors okay first start with
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00:41:24,780 --> 00:41:38,380
two dimensions if i give you any 2 witness
let us call them v1 and v2 are these 2 vectors
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00:41:38,380 --> 00:41:52,180
in two dimensions what is this set so span
of v1 v2 this corresponds to alpha v1 plus
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00:41:52,180 --> 00:42:04,910
beta v2 for any alpha beta that belong to
f okay that belong to the field f
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00:42:04,910 --> 00:42:11,609
what is this set field f is a real numbers
what is this set it is a plane passing through
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00:42:11,609 --> 00:42:20,089
origin okay because two independent vectors
two linearly independent vectors you have
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00:42:20,089 --> 00:42:24,320
to defined what is linear independence so
2 linearly independent vectors if i take all
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00:42:24,320 --> 00:42:31,829
possible linear combination then what i get
is the span and the span is nothing but yeah
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00:42:31,829 --> 00:42:40,769
well let us say that this is a subspace this
is the subspace if i give you a third vector
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00:42:40,769 --> 00:42:49,920
in the same subspace okay which is v3 okay
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00:42:49,920 --> 00:43:01,749
what will be alpha v1 plus beta v2 plus gamma
v3 okay that is because this third vector
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00:43:01,749 --> 00:43:12,989
v3 is linearly depend upon v2 and then you
can and what you get here so if i have some
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00:43:12,989 --> 00:43:19,170
more vectors belonging to the same set here
okay same subspace and if i take all possible
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00:43:19,170 --> 00:43:25,069
linear combinations i am not going to get
a different set i am not going to get a different
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00:43:25,069 --> 00:43:28,180
set okay i will get the same set which is
this plane i cannot leave this plane
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00:43:28,180 --> 00:43:32,089
if i take linear combination of any two vectors
in this plane i cannot leave this plane what
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00:43:32,089 --> 00:43:41,829
is the minimum number of vectors that are
required to generate this thing okay so the
323
00:43:41,829 --> 00:43:47,390
minimum number of vectors and are required
to generate a subspace or space is called
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00:43:47,390 --> 00:43:53,250
as the dimension of the vector space okay
what is the dimension of this particular subspace
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00:43:53,250 --> 00:43:59,150
this is a subspace right the span is a subspace
what is the dimension dimension is 2
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00:43:59,150 --> 00:44:08,920
because you need only 2 2 linearly independent
vectors to generate all vectors inside this
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00:44:08,920 --> 00:44:16,460
plane which is passing through the origin
okay this is a 2 dimensional subspace of it
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00:44:16,460 --> 00:44:33,710
is the two dimensional subspace of everyone
will know this okay now let us consider these
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00:44:33,710 --> 00:44:47,269
2 vectors what will be span of these 2 vectors
professor student conversation starts correct
330
00:44:47,269 --> 00:44:49,579
what dimensional okay
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00:44:49,579 --> 00:45:10,819
so if i take span of v1 v2 it is alpha times
1 2 3 4 and 5 plus beta times 5 4 3 2 1 for
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00:45:10,819 --> 00:45:26,690
any alpha beta belongs to r this all possible
linear combinations of these two vectors is
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00:45:26,690 --> 00:45:37,220
called a span of these two vectors and this
span will be nothing but a two dimensional
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00:45:37,220 --> 00:45:50,690
subspace of r5 what is r5 space considering
of vectors each vector has 5 doubles yes 5
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00:45:50,690 --> 00:46:01,369
components so number of components in the
vector does not define the dimension okay
336
00:46:01,369 --> 00:46:05,079
this is the fifth 5 dimensional couple
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00:46:05,079 --> 00:46:11,109
just because it has 5 components does not
mean this linear combination will define if
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00:46:11,109 --> 00:46:19,789
i take only one vector say v1 what will be
alpha times v1 it is a line and it is one
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00:46:19,789 --> 00:46:22,800
dimensional subspace it is a one dimensional
subspace of r5 5 dimensional space okay may
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00:46:22,800 --> 00:46:25,829
be one dimensional subspace of r5professor
student conversation ends well so far so good
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00:46:25,829 --> 00:46:35,259
we follow and define what is basis and who
want some more insights into why this is all
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00:46:35,259 --> 00:46:37,789
required where do i need this we will do in
our next lecture