Linear Algebra: Basis of a Subspace

Understanding the definition of a basis of a subspace

Let's say I have
the subspace v.

And this is a subspace and we
learned all about subspaces in

the last video.

And it's equal to the span
of some set of vectors.

And I showed in that video that
the span of any set of

vectors is a valid subspace.

It's going to be the span of v1,
v2, all the way, so it's

going to be n vectors.

So each of these are vectors.

Now let me also say that all of
these vectors are linearly

independent.

So v1, v2, all the way to vn,
this set of vectors are

linearly independent.

Now before I kind of give you
the punchline, let's review

what exactly span meant.

Span meant that this set, this
subspace, represents all of

the possible linear
combinations of

all of these vectors.

So you know, I could have all of
the combinations for all of

the different c's.

So c1 times v1 plus c2 times v2,
all the way to cn times vn

for all of the possible c's
and the real numbers.

If you take all of the
possibilities of these and you

put all of those vectors into
a set, that is the span and

that's what we're defining
the subspace v as.

Now, the definition of linear
independence meant that the

only solution to c1, v1, plus
c2, v2 plus all the way to cn,

vn, that the only solution to
this equally the 0 vector--

maybe I should put a little
vector sign up there-- is when

all of these terms
are equal to 0.

c1 is equal to c2, is equal
to all of these.

All of them are equal to 0.

Or kind of a more common sense
way to think of it is that you

can't represent any one of these
vectors as a combination

of the other vectors.

Now, if both of these conditions
are true that the

span of this set of vectors is
equal to this subspace or

creates this subspace or it
spans this subspace, and that

all of these vectors are
linearly independent, then we

can say that the set of
vectors-- maybe we call this,

we could call this
set of vectors s.

Where we say s is equal to v1,
v2, all the way to vn.

It's equal to that
set of vectors.

We can then say and this
is the punchline.

We can then say that S, the
set S is a basis for v.

And this is the definition
I wanted to make.

If something is a basis for a
set, that means that those

vectors, if you take the span
of those vectors, you can

construct-- you can get to any
of the factors in that

subspace and that those
vectors are linearly

independent.

So there's a couple of ways
to think about it.

One is there's a lot of things
that might span for something.

For example, if this spans for
v, then so would-- let me add

another vector.

Let me define another set.

Let me define set T to be
all of set S: v1, v2,

all the way to vn.

But it also contains
this other vector.

I'm going to call it the
v special vector.

So it's going to be essentially,
the set S plus

one more vector.

Where this vector I'm just
saying is equal to v1 plus v2.

So clearly, this is not a
linearly independent set.

But if I had asked you what the
span of T is, the span of

T is still going to be
this subspace, v.

But I have this extra vector
in here that made it

non-linearly independent.

This set is not linearly
independent.

So T is linearly dependent.

So in this case, T is
not a basis for v.

And I had showed you this
example because the way my

head thinks about basis is,
the basis is really the

minimum set of vectors that I
need, the minimum set-- and

I'll write this down.

This isn't a formal definition,
but I view a

basis-- let me switch colors--
as really the-- let me get a

good color here.

As a basis is the minimum-- I'll
put it in quotes because

I haven't defined that.

The minimum set of vectors that
spans the space that it's

a basis of, spans
the subspace.

So in this case, this is the
minimum set of vectors.

And I'm not going to prove it
just yet, but you can see

that, look.

This set of vectors right
here, it does span the

subspace, but it's clearly not
the minimum set of vectors.

Because the span of this thing,
I could still remove

this last vector here.

I could still remove that guy
and still-- and then the span

of what's left over is still
going to span my subspace v.

So this guy right here
was redundant.

In a basis, you have
a no redundancy.

Each one of these guys is needed
to be able to construct

any of the vectors in
the subspace v.

Let me do some examples.

So let's just take some
vectors here.

Let's say I had to find
my set of vectors, and

I'll deal in r2.

So let's say I have
the vector 2, 3.

Let's say I have the
other vector 7, 0.

So first of all, let's just
think about the span, the span

of this set of vectors.

This is a set of vectors.

So what's the span of S?

What's all of the linear
combinations of this?

Well, let's see if
it's all of r2.

So if it's all of r2 that means
the linear combination

of this could be-- we could
always construct anything in

r2 with the linear combination
of this.

So if we have c1 times 2,
3 plus c2 times 7, 0.

If it is true that this spans
all of r2, then we should be

able to construct-- we should
always be able find a c1 and a

c2 to construct any
point in r2.

And let's see if we
can show that.

So we get 2c1 plus 7c2
is equal to x1.

And then we get 3c1 plus 0c2.

Plus 0 is equal to x2.

And if we take this second
equation and divide both sides

by 3 we get c1 is equal
to x2 over 3.

And then if we substitute that
back into this first equation,

we get 2/3-- I'm just
substituting c1 in there.

So 2/3 x2.

2 times x2 over 3 is 2/3 x2.
plus 7c2 is equal to x1.

And then, what can we do?

We can subtract the 2/3
x2 from both sides.

Let me do it right here.

So we get 7c2 is equal
to x1 minus 2/3 x2.

Divide both sides of this
by 7 and you get c2.

Let me do it in yellow.

You get c2 is equal to x1 over
7 minus 2 over 21 x2.

So if you've given me any x1 and
any x2 where either x1 or

x2 are a member of the real
numbers, we're talking about--

well, everything we're going to
be dealing with right now

is real numbers.

You give me any two
real numbers.

I take my x2 divided by 3 and
I'll give you your c1.

And I'll take the x1 divided
by 7 and subtract

2/21 times your x2.

And I'll get you your c2.

This will never break.

There's no division
by any of these.

You don't have to worry about
these equaling a 0.

These two formulas
will always work.

So you give me any x1 and any
x2, I can always find

you a c1 or a c2.

Which is essentially finding a
linear combination that will

equal your vector.

So the span of S is r2.

Now the second question is, is
are these two vectors linearly

independent?

And linear independence means
that the only solution to the

equation c-- let me
switch colors.

The only solution to the
equation c1 times the first

vector plus c2 times the second
vector equaling the 0

vector, that the only solution
to this is when both

of these equal 0.

So let's see if that's true.

We've already solved for it, so
if x1-- in this case, x1 is

equal to 0 and x2
is equal to 0.

This is just a special
case where I'm making

them equal to 0 vector.

If I want to get the 0 vector,
c1 is equal to 0/3.

So c1 must be equal to 0.

And c2 is equal to 0/7
minus 2/21 times 0.

So c2 must also be equal to 0.

So the only solution to this
was settings both of these

guys equal to 0.

So S is also a linearly
independent set.

So it spans r2, it's linearly
independent.

So we can say definitively, that
S-- that the set S, the

set of vectors S is
a basis for r2.

Now, is this the only
basis for r2?

Well I could draw a trivially
simple vector, set of vectors.

I could do this one.

Let me call it T.

If I define T to be the set 1, 0
and 0, 1, does this span r2?

Let's say I want to generate
the-- I want to

get to x1 and x2.

How can I construct that out
of these two vectors?

Well, if I always just do x1
times 0, 1 plus x2 times 0, 1,

that'll always give me x1, x2.

So this definitely
does span r2.

Is it linearly independent?

I could show it to you.

If you wanted to make this
equal to the 0 vector.

If this is a 0 and this is 0,
then this has to be a 0 and

this has to be a 0.

And that's kind of obvious.

There's no way that you could
get one of the other vectors

by some multiple of
the other one.

There's no way you could get a
1 here by multiplying this by

anything and vice versa.

So it's also linearly
independent.

And the whole reason why I
showed you this is because I

wanted to show you that look,
this set T spans r2.

It's also linearly independent,
so T is also a

basis for r2.

And I wanted to show you this
to show that if I look at a

vector subspace and r2 is a
valid subspace of itself.

You can verify that.

But if I have a subspace, it
doesn't have just one basis.

It could have multiple bases.

In fact, it normally
has infinite bases.

So in this case, S is a valid
basis and T is also a valid

basis for r2.

And actually, just so you know
what T is, the situation here,

this is called a
standard basis.

This is the standard basis.

And this is what you're used
to dealing with in just

regular calculus or
physics class.

And if you remember from physics
class, this is the

unit vector i and then this
is the unit vector j.

And it's the standard basis for
two-dimensional Cartesian

coordinates.

What's useful about a basis is
that you can always-- and it's

not just true of the standard
basis, is that you can

represent any vector
in your subspace.

You can represent any vector
in your subspace by some

unique combination of the
vectors in your basis.

So let me show you that.

So let's say that the set v1,
v2, all the way to vn, let's

say that this is a basis for--
I don't know-- just some

subspace U.

So this is a subspace.

So that means that these guys
are linearly independent.

And also means that the span of
these guys, or all of the

linear combinations of these
vectors, will get you all of

the vectors, all of the possible
components, all of

the difference members of U.

Now what I want to show you is
each member of U can only be

uniquely defined by a unique set
of-- a unique combination

of these guys.

Let me be clear about that.

Let's say my vector a is a
member of our subspace U.

That means that a can be
represented by some linear

combination of these guys.

These guys span U.

So that means that we can
represent our vector a as

being c1 times v1 plus
c2 times v2.

These are vectors.

All the way to cn times vn.

Now I want to show you that this
is a unique combination.

And to show that I'm going to
prove by contradiction.

Let's say that there's
another combination.

Let's say I could also represent
a by some other

combination, d1 times v2 plus
d2 times v2 plus all the way

to dn times vn.

Now, what happens if I
subtract a from a?

I'm going to get the 0 vector.

Let me subtract these
two things.

If I subtract a from
a, a minus a is

clearly the 0 vector.

It's clearly the 0 vector and
if I subtract this side from

that side, what do we get?

I'll do it in a different
color.

We get c1 minus d1 times v1 plus
c2 minus d2 times v2, all

the way to-- I'm at the point on
my board where it starts to

malfunction.

All the way to c-- you can't
see it. cn minus vn.

It's showing up somehow.

cn minus-- no, it's
messing me up.

Let me rewrite it on the left
right here where it's less

likely to mess up.

The 0 vector, I'll write
it like that.

Is equal to c1 minus d1 times
v1 plus all the way up to cn

minus dn times vn.

I just subtracted the
vector by itself.

Now, I told you that
these are a basis.

There's two things that-- when
you say a basis, it says that

the span of these guys
makes the subspace.

Or the span of these guys
is the subspace.

And it also tells you that
these guys are linearly

independent.

So if they're linearly
independent, the only solution

to this equation-- this is just
a constant times v1 plus

another constant times
v2, all the way to a

constant times vn.

The only solution to this
equation is if each of these

constants equal 0.

So all of those constants
have to be equal to 0.

Over here before it messed up,
this has to equal 0, this has

to equal 0.

That was a definition of
linear independence.

And we know that this is a
linearly independent set.

So if all of those constants are
equal to 0, then we know

that c1-- if this is equal to 0,
then c1 is equal to d1, c2

is equal to d2, all the way
to cn is equal to dn.

So by the fact that it's
linearly independent, all of

these-- each of these constants
have to be equal to

each other.

And that's our contradiction.

I assume they're different, but
the linear independence

forced them to be the same.

So if you have a basis for some
subspace, any member of

that subspace can be uniquely
determined by a unique

combination of those vectors.

And just to hit the point home,
I told you that this was

a basis for r2.

And my next question is,
and I just want to kind

of backtrack a bit.

If I just added another vector
here, if I just added the

vector 1, 0, is S now
a basis for r2?

Well no, it clearly will
continue to span r2, but this

guy is redundant.

This guy is in r2.

And I already told you that
these two guys alone span r2.

That anything in r2 can be
represented by a linear

combination of these two guys.

This guy is clearly in r2, so
he can be represented by a

linear combination of
these two guys.

So therefore, this is not a
linearly independent set.

This is linearly dependent.

And because it's literally
dependent, I have redundant

information here.

And then this would no
longer be a basis.

So in order for these to be a
basis I kind of have to create

the minimum set of vectors that
can span, or the most

efficient set of vectors that
can span, in this case, r2.