let's review our notions of subspaces
again and then let's see if we can
define some interesting subspaces
dealing with matrices and vectors so a
subspace let me--just subspace let's say
that I have some subspace oh let me just
call it some subspace s this is a
subspace if the following are true and
this is all a review that the zero
vector I'll just do it like that the
zero vector is a member of s what
contains the zero vector that if v1 and
v2 are both members of my subspace then
v1 plus v2 is also a member of my
subspace so that's just saying that the
subspaces are closed under addition you
can add any of their two members and
you'll get another member of the
subspace and then the last requirement
if you remember is that subspaces are
closed under multiplication so that if C
is a real number and it's just a scalar
and if I multiply and v1 is a member of
my subspace then if I multiply that
arbitrary real number times my member of
my subspace v1 I'm gonna get another
member of the subspace so it's closed
under multiplication these were all of
what a subspace is you know there's our
definition of the subspace if you call
something a subspace these need to be
true
now let's see if we can do something
interesting with what we understand
about matrix vector multiplication let's
say I have the vector matrix a the
matrix a I'm making nice and bold and
it's an M by n matrix and I'm interested
in the following situation I want to set
up the homogenious
equation and we'll talk about why it's
homogeneous well I'll tell you in a
second so let's say we set up the
equation my matrix a times vector X is
equal to the zero vector is equal to the
zero vector this is a homogeneous
equation because we have a 0 there homo
genius and I want to ask the question
I've talked about subspaces is if I take
all of the X's
if I take the world the universe the set
of all of the X's that satisfy this
equation do I have a valid subspace so
if I take let's think about this I want
to take all of the X's that are a member
of our and remember if if our matrix a
has n columns that I've only defined
this matrix vector multiplication if X
is a member of our and if X has to have
exactly n components only then is it
defined so let me define a set of all of
the vectors that are a member of RN
where they satisfy the equation a times
my vector X is equal to the zero vector
so my question is is this a subspace is
this a valid subspace so the first
question is does it contain the zero
vector well in order for this to contain
the zero vector the zero vector must
satisfy this equation so what is any M
by n matrix a times the zero vector I'll
write it like that times the zero vector
well let's write out my matrix a my
matrix a a 1 1 a 1 2 all the way to a 1n
and then this as we go down a column we
go all the way down to a m1 and then as
we go all the way to the bottom right we
go to a M N and I'm going to multiply
that times a unit vector I'm sorry I'm
gonna multiply that times the zero
vector that has exactly n components so
the zero vector with n components is 0 0
and you're gonna have n of these the
number of components here has to be
exact same number of the number of
columns you have but when you take this
product this matrix vector product what
do you get what do we get well this
first term up here is going to be a 1 1
times 0 plus a 1 2 times 0 plus each of
these terms times 0 so then you add them
all up a 1 1 times 0 plus a 1 2 times 0
all the way to a 1 n times this 0 so you
get 0 now this term is going to be a 2 1
times 0 plus a
- two times zero plus a23 times zero all
the way to a to n times zero that's
obviously going to be zero and you're
going to keep doing that because all of
these are essentially you can kind of
view it as the dot product of well if
you want if I haven't defined dot
product with row vectors in column
vectors but I think you get the idea the
prot the sum of each of these elements
multiplied with the corresponding
component in this vector and of course
you're just always multiplying by zero
and then adding up so you're going to
get nothing but a bunch of zeros so the
zero vector does satisfy the equation a
times the zero vector is equal to the
zero vector and this is a very
unconventional notation I'm just writing
it like that cuz I don't feel like
bolding out my zeros all the time to
make you realize that that's a vector so
we we meet our first requirement the
zero vector is a member of this set so
it does contain so let me define my set
here let me define it n and I'll tell
you in a second while I'm calling it n
so we now know that the zero vector the
zero vector is a member of my set n now
let's say I have two vectors V 1 and V 2
that are that set that our members let
me write this so let's say I have two
vectors V 1 and V 2 and this was a V 2
over here V 1 and V 2 that are both
members of our set what does that mean
that means that they both satisfy this
equation so that means that means that a
my matrix a times vector one is equal to
zero this is by definition I'm saying
that they're member of this set which
means they must satisfy this and that
also means that a times vector two is
equal to our zero vector so in order for
this to be closed under addition a times
a times vector one plus vector 2 the sum
of these two vectors should also be a
member of n but let's say figure out
what this is the sum of these two
vectors is this vector right here this
is equal to and I haven't proven this to
you yet I haven't made a video where I
approve this but it's very easy to prove
just using the definition of matrix
vector multiplication that matrix vector
multiplication does display the
distributive property
maybe I'll make a video on that but it's
literally you just have to go through
the mechanics of each of the terms this
is equal to a v1 plus a v2 and we know
that this is equal to the zero vector
and this is equal to the zero vector
and if you add the zero vector to itself
this whole thing is going to be equal to
the zero vector
so if v1 is a member of N and v2 is a
member of n which means they both
satisfy this equation then v1 plus v2 is
definitely still a member of n because
when I multiply a times that I get the
zero vector again so we get let me write
that result as well so we are also let
me write that so we are also let me
write this right here so we now know
that v1 plus v2 is also a member of N
and the last thing we have to show is
that it's closed under multiplication so
let's say that v1 is a member of our
space that I defined here where they
satisfy this equation what about what
about C times v1 what about C times v1
is that a member of n well let's think
about it what's a our vector our matrix
a time's the vector right I'm just
multiplying this times a scared I'm
gonna get another vector the I don't
wanna write a capital V they're
lowercase V so it's a vector so what's
this equal to well once again I haven't
proven it to you yet but it's actually a
very straightforward thing to do to show
that that the the if when you're dealing
with scalars if you have a scalar here
it doesn't matter if you multiply the
scalar times a vector before multiplying
it times the the the matrix or
multiplying the matrix times a vector
and then doing the scalar so it's very
it's straight fairly straightforward to
prove that this is equal to C times our
matrix a I'll make that nice and bold
times our vector V that these two things
are equivalent I haven't maybe I should
just turn out the video that does this
but I'll leave it to you it's very you
literally just go through the mechanics
by component by component you show this
but clearly if this is true if this is
true we already know that v1
sorry this was v1 we already know that
v1 is a member of our set which means
that a times v1 is equal to the zero
vector and so that means well this will
reduce to C times the zero vector which
is still the zero vector so C v1 is
definitely a member of n so it's closed
under multiplication and you know I kind
of assumed this right here but maybe
I'll prove it in a different video but I
want to do all this to show that this
set n is a valid subspace this is a
valid subspace that contains the zero
vector it's closed under addition it's
closed under multiplication and we
actually have a special name for this we
call this right here we call n we call n
the null space the null space the null
space of a or we could write n is equal
to maybe I shouldn't of written an end
written orange in there our orange n is
equal to the notation is just the null
space of a or we could write the null
space is equal the orange notation of N
and so literally if I just give you some
arbitrary matrix a and I say hey find me
and of a what is that you literally
you're you are your goal is to find the
set of all X's the set of all X's that
satisfy the equation a times X is equal
to 0 and I'm going to do that in the
next video