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# Introduction to linear independence

Video transcript

Let's say I had the set of
vectors-- I don't want to do it that thick. Let's say one of the vectors is
the vector 2, 3, and then the other vector is
the vector 4, 6. And I just want to answer the
question: what is the span of these vectors? And let's assume that these
are position vectors. What are all of the
vectors that these two vectors can represent? Well, if you just look at it,
and remember, the span is just all of the vectors that can
be represented by linear combinations of these. So it's the set of all the
vectors that if I have some constant times 2 times that
vector plus some other constant times this vector,
it's all the possibilities that I can represent when I just
put a bunch of different real numbers for c1 and c2. Now, the first thing you might
realize is that, look, this vector 2, this is just
the same thing as 2 times this vector. So I could just rewrite
it like this. I could just rewrite it as c1
times the vector 2, 3 plus c2 times the vector-- and here,
instead of writing the vector 4, 6, I'm going to write 2 times
the vector 2, 3, because this vector is just a multiple
of that vector. So I could write c2 times
2 times 2, 3. I think you see that this is
equivalent to the 4, 6. 2 times 2 is 4. 2 times 3 is 6. Well, then we can simplify
this a little bit. We can rewrite this as just c1
plus 2c2, all of that, times 2, 3, times our vector 2, 3. And this is just some
arbitrary constant. It's some arbitrary constant
plus 2 times some other arbitrary constant. So we can just call this c3
times my vector 2, 3. So in this situation, even
though we started with two vectors, and I said, well, you
know, the span of these two vectors is equal to all of
the vectors that can be constructed with some linear
combination of these, any linear combination of these,
if I just use this substitution right here, can
be reduced to just a scalar multiple of my first vector. And I could have gone the
other way around. I could have substituted this
vector as being 1/2 times this, and just made any
combination of scalar multiple of the second vector. But the fact is, that instead
of talking about linear combinations of two vectors,
I can reduce this to just a scalar combination
of one vector. And we've seen in R2 a scalar
combination of one vector, especially if they're
position vectors. For example, this vector 2, 3. It's 2, 3. It looks like this. All the scalar combinations of
that vector are just going to lie along this line. So 2, 3, it's going
to be right there. They're all just going to lie
along that line right there, so along this line going in
both directions forever. And if I take negative
values of 2, 3, I'm going to go down here. If I take positive values,
I'm going to go here. If I get really large positive
values, it's going to go up here. But I can just represent the
vectors, and when you put them in standard form, their arrows
essentially would trace out this line. So you could say that the span
of my set of vectors-- let me put it over here. The span of the set of vectors
2, 3 and 4, 6 is just this line here. Even though we have
two vectors, they're essentially collinear. They're multiples
of each other. I mean, if this is 2, 3, 4,
6 is just this right here. It's just that longer
one right there. They're collinear. These two things
are collinear. Now, in this case, when we have
two collinear vectors in R2, essentially their span just
reduces to that line. You can't represent
some vector like-- let me do a new color. You can't represent this vector
right there with some combination of those
two vectors. There's no way to kind of
break out of this line. So there's no way that you can
represent everything in R2. So the span is just
that line there. Now, a related idea to this,
and notice, you had two vectors, but it kind of reduced
to one vector when you took its linear combinations. The related idea here is that we
call this set-- we call it linearly dependent. Let me write that down:
linearly dependent. This is a linearly
dependent set. And linearly dependent just
means that one of the vectors in the set can be represented
by some combination of the other vectors in the set. A way to think about it is
whichever vector you pick that can be represented by the
others, it's not adding any new directionality or any
new information, right? In this case, we already had
a vector that went in this direction, and when you throw
this 4, 6 on there, you're going in the same direction,
just scaled up. So it's not giving us any new
dimension, letting us break out of this line, right? And you can imagine in three
space, if you have one vector that looks like this and another
vector that looks like this, two vectors that aren't
collinear, they're going to define a kind of two-dimensional
space. They can define a
two-dimensional space. Let's say that this is
the plane defined by those two vectors. In order to define R3, a third
vector in that set can't be coplanar with those
two, right? If this third vector is coplanar
with these, it's not adding any more directionality. So this set of three
vectors will also be linearly dependent. And another way to think about
it is that these two purple vectors span this plane, span
the plane that they define, essentially, right? Anything in this plane going in
any direction can be-- any vector in this plane, when we
say span it, that means that any vector can be represented
by a linear combination of this vector and this vector,
which means that if this vector is on that plane, it can
be represented as a linear combination of that vector
and that vector. So this green vector I added
isn't going to add anything to the span of our set of vectors
and that's because this is a linearly dependent set. This one can be represented by a
sum of that one and that one because this one and this
one span this plane. In order for the span of these
three vectors to kind of get more dimensionality or start
representing R3, the third vector will have to break
out of that plane. It would have to break
out of that plane. And if a vector is breaking out
of that plane, that means it's a vector that can't be
represented anywhere on that plane, so it's outside of the
span of those two vectors. Where it's outside, it can't
be represented by a linear combination of this
one and this one. So if you had a vector of this
one, this one, and this one, and just those three, none of
these other things that I drew, that would be linearly
independent. Let me draw a couple more
examples for you. That one might have been
a little too abstract. So, for example, if I have the
vectors 2, 3 and I have the vector 7, 2, and I have the
vector 9, 5, and I were to ask you, are these linearly
dependent or independent? So at first you say, well, you
know, it's not trivial. Let's see, this isn't a scalar
multiple of that. That doesn't look like a scalar
multiple of either of the other two. Maybe they're linearly
independent. But then, if you kind of inspect
them, you kind of see that v, if we call this v1,
vector 1, plus vector 2, if we call this vector 2, is
equal to vector 3. So vector 3 is a linear
combination of these other two vectors. So this is a linearly
dependent set. And if we were to show it, draw
it in kind of two space, and it's just a general idea
that-- well, let me see. Let me draw it in R2. There's a general idea that if
you have three two-dimensional vectors, one of them is
going to be redundant. Well, one of them definitely
will be redundant. For example, if we do 2, 3, if
we do the vector 2, 3, that's the first one right there. I draw it in the standard
position. And I draw the vector 7, 2 right
there, I could show you that any point in R2 can be
represented by some linear combination of these
two vectors. We can even do a kind of a
graphical representation. I've done that in the previous
video, so I could write that the span of v1 and v2
is equal to R2. That means that every vector,
every position here can be represented by some linear
combination of these two guys. Now, the vector 9,
5, it is in R2. It is in R2, right? Clearly. I just graphed it
on this plane. It's in our two-dimensional,
real number space. Or I guess we could call it
a space or in our set R2. It's there. It's right there. So we just said that anything in
R2 can be represented by a linear combination of
those two guys. So clearly, this is in R2, so
it can be represented as a linear combination. So hopefully, you're starting
to see the relationship between span and linear
independence or linear dependence. Let me do another example. Let's say I have the vectors--
let me do a new color. Let's say I have the vector--
and this one will be a little bit obvious-- 7, 0, so that's
my v1, and then I have my second vector, which
is 0, minus 1. That's v2. Now, is this set linearly
independent? Is it linearly independent? Well, can I represent
either of these as a combination of the other? And really when I say as a
combination, you'd have to scale up one to get the other,
because there's only two vectors here. If I am trying to add up to this
vector, the only thing I have to deal with is this
one, so all I can do is scale it up. Well, there's nothing
I can do. No matter what I multiply this
vector by, you know, some constant and add it to itself
or scale it up, this term right here is always
going to be zero. It's always going to be zero. So nothing I can multiply
this by is going to get me to this vector. Likewise, no matter what I
multiply this vector by, the top term is always
going to be zero. So there's no way I could
get to this vector. So both of these vectors,
there's no way that you can represent one as a combination
of the other. So these two are linearly
independent. And you can even see
it if we graph it. One is 7, 0, which
is like that. Let me do it in a non-yellow
color. 7, 0. And one is 0, minus 1. And I think you can clearly see
that if you take a linear combination of any of these
two, you can represent anything in R2. So the span of these, just to
kind of get used to our notion of span of v1 and v2,
is equal to R2. Now, this is another interesting
point to make. I said the span of
v1 and v2 is R2. Now what is the span of
v1, v2, and v3 in this example up here? I already told you. I already showed you that
this third vector can be represented as a linear
combination of these two. It's actually just these
two summed up. I can even draw it right here. It's just those two
vectors summed up. So it clearly can be represented
as a linear combination of those two. So what's its span? Well, the fact that this is
redundant means that it doesn't change its span. It doesn't change all of the
possible linear combinations. So its span is also
going to be R2. It's just that this was
more vectors than you needed to span R2. R2 is a two-dimensional space,
and you needed two vectors. So this was kind of a more
efficient way of providing a basis, and I haven't defined
basis formally, yet, but I just want to use it a little
conversationally, and then it'll make sense to you when
I define it formally. This provides a better basis, or
this provides a basis, kind of a non-redundant set of
vectors that can represent R2. While this one, right
here, is redundant. So it's not a good
basis for R2. Let me give you one more example
in three dimensions. And then in the next video,
I'm going to make a more formal definition of linear
dependence or independence. So let's say that I had
the vector 2, 0, 0. Let me make a similar argument
that I made up there: the vector 2, 0, 0, the vector 0, 1,
0, and the vector 0, 0, 7. We are now in R3, right? Each of these are
three-dimensional vectors. Now, are these linear dependent
or linearly independent? Sorry, are they linearly
dependent or independent? Well, there's no way with some
combination of these two vectors that I can end up with
a non-zero term right here to make this third vector, right? Because no matter what I
multiply this one by and this one by, this last term
is going to be zero. So this is kind of adding
a new direction to our set of vectors. Likewise, there's nothing
I can do-- there's no combination of this guy and
this guy that I can get a non-zero term here. And finally, no combination of
this guy and this guy that I can get a non-zero term here. So this set is linearly
independent. And if you were to graph these
in three dimensions, you would see that none of these--
these three do not lie on the same plane. Obviously, any two of them lie
on the same plane, but if you were to actually graph
it, you get 2, 0. Let me say that that's x-axis. That's 2, 0, 0. Then you have this, 0, 1, 0. Maybe that's the y-axis. And then you have 0, 0, 7. It would look something
like this. So it almost looks like, your
three-dimensional axes, it almost looks like the
vectors i, j, k. They're just scaled
up a little bit. But you can always correct
that by just scaling them down, right? Because we care about any linear
combination of these. So the span of these three
vectors right here, because they're all adding new
directionality, is R3. Anyway, I thought I would leave
you there in this video. I realize I've been making
longer and longer videos, and I want to get back in the habit
of making shorter ones. In the next video, I'm going
to make a more formal definition of linear dependence,
and we'll do a bunch more examples.