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# Linear combinations and span

Understanding linear combinations and spans of vectors. Created by Sal Khan.

Video transcript

One term you are going to hear
a lot of in these videos, and in linear algebra in general,
is the idea of a linear combination. And all a linear combination of
vectors are, they're just a linear combination. Let me show you what
that means. So let's say I have a couple
of vectors, v1, v2, and it goes all the way to vn. And they're all in, you know,
it can be in R2 or Rn. Let's say that they're
all in Rn. They're in some dimension of
real space, I guess you could call it, but the idea
is fairly simple. A linear combination of these
vectors means you just add up the vectors. It's some combination of a sum
of the vectors, so v1 plus v2 plus all the way to vn,
but you scale them by arbitrary constants. So you scale them by c1, c2,
all the way to cn, where everything from c1
to cn are all a member of the real numbers. That's all a linear
combination is. Let me show you a concrete
example of linear combinations. Let me make the vector. Let me define the vector a to
be equal to-- and these are all bolded. These purple, these are all
bolded, just because those are vectors, but sometimes it's
kind of onerous to keep bolding things. So let's just say I define the
vector a to be equal to 1, 2. And I define the vector
b to be equal to 0, 3. What is the linear combination
of a and b? Well, it could be any constant
times a plus any constant times b. So it could be 0 times a plus--
well, it could be 0 times a plus 0 times b, which,
of course, would be what? That would be 0 times 0,
that would be 0, 0. That would be the 0 vector, but
this is a completely valid linear combination. And we can denote the
0 vector by just a big bold 0 like that. I could do 3 times a. I'm just picking these
numbers at random. 3 times a plus-- let me do a
negative number just for fun. So I'm going to do plus
minus 2 times b. What is that equal to? Let's figure it out. Let me write it out. It's 3 minus 2 times 0,
so minus 0, and it's 3 times 2 is 6. 6 minus 2 times 3, so minus 6,
so it's the vector 3, 0. This is a linear combination
of a and b. I can keep putting in a bunch
of random real numbers here and here, and I'll just get a
bunch of different linear combinations of my
vectors a and b. If I had a third vector here,
if I had vector c, and maybe that was just, you know, 7, 2,
then I could add that to the mix and I could throw in
plus 8 times vector c. These are all just linear
combinations. Now why do we just call
them combinations? Why do you have to add that
little linear prefix there? Because we're just
scaling them up. We're not multiplying the
vectors times each other. We haven't even defined what it
means to multiply a vector, and there's actually several
ways to do it. But, you know, we can't square
a vector, and we haven't even defined what this means yet, but
this would all of a sudden make it nonlinear
in some form. So all we're doing is we're
adding the vectors, and we're just scaling them up by some
scaling factor, so that's why it's called a linear
combination. Now you might say, hey Sal, why
are you even introducing this idea of a linear
combination? Because I want to introduce the
idea, and this is an idea that confounds most students
when it's first taught. I think it's just the very
nature that it's taught. Over here, I just kept putting
different numbers for the weights, I guess we could call
them, for c1 and c2 in this combination of a and b, right? Let's ignore c for
a little bit. I just put in a bunch of
different numbers there. But it begs the question: what
is the set of all of the vectors I could have created? And this is just one
member of that set. But what is the set of all of
the vectors I could've created by taking linear combinations
of a and b? So let me draw a and b here. Maybe we can think about it
visually, and then maybe we can think about it
mathematically. So let's say a and b. So a is 1, 2. So 1, 2 looks like that. That's vector a. Let me do vector b in
a different color. We're going to do
it in yellow. Vector b is 0, 3. So vector b looks
like that: 0, 3. So what's the set of all of
the vectors that I can represent by adding and
subtracting these vectors? And we said, if we multiply them
both by zero and add them to each other, we
end up there. If we take 3 times a, that's
the equivalent of scaling up a by 3. So you go 1a, 2a, 3a. So that's 3a, 3 times a
will look like that. So this vector is 3a, and then
we added to that 2b, right? Oh no, we subtracted 2b
from that, so minus b looks like this. Minus 2b looks like this. This is minus 2b, all the way,
in standard form, standard position, minus 2b. So if you add 3a to minus 2b,
we get to this vector. 3a to minus 2b, you get this
vector right here, and that's exactly what we did when we
solved it mathematically. You get the vector 3, 0. You get this vector
right here, 3, 0. But this is just one
combination, one linear combination of a and b. Instead of multiplying a times
3, I could have multiplied a times 1 and 1/2 and just
gotten right here. So 1 and 1/2 a minus 2b would
still look the same. It would look like something
like this. It would look something like--
let me make sure I'm doing this-- it would look something
like this. And so our new vector that
we would find would be something like this. So I just showed you, I can find
this vector with a linear combination. I can find this vector with
a linear combination. And actually, it turns out that
you can represent any vector in R2 with some linear
combination of these vectors right here, a and b. Now, let's just think of an
example, or maybe just try a mental visual example. Wherever we want to go, we
could go arbitrarily-- we could scale a up by some
arbitrary value. So this is some weight on a,
and then we can add up arbitrary multiples of b. B goes straight up and down,
so we can add up arbitrary multiples of b to that. So we could get any point on
this line right there. Now, if we scaled a up a little
bit more, and then added any multiple b, we'd get
anything on that line. If we multiplied a times a
negative number and then added a b in either direction, we'll
get anything on that line. We can keep doing that. And there's no reason why we
can't pick an arbitrary a that can fill in any of these gaps. If we want a point here, we just
take a little smaller a, and then we can add all
the b's that fill up all of that line. So we can fill up any
point in R2 with the combinations of a and b. So what we can write here is
that the span-- let me write this word down. The span of the vectors a and
b-- so let me write that down-- it equals R2 or it equals
all the vectors in R2, which is, you know, it's
all the tuples. R2 is all the tuples
made of two ordered tuples of two real numbers. So it equals all of R2. This just means that I can
represent any vector in R2 with some linear combination
of a and b. And you're like, hey, can't I do
that with any two vectors? Well, what if a and b were the
vector-- let's say the vector 2, 2 was a, so a is equal to 2,
2, and let's say that b is the vector minus 2, minus
2, so b is that vector. So b is the vector
minus 2, minus 2. Now, can I represent any
vector with these? Well, I can scale a up and down,
so I can scale a up and down to get anywhere on this
line, and then I can add b anywhere to it, and
b is essentially going in the same direction. It's just in the opposite
direction, but I can multiply it by a negative and go
anywhere on the line. So any combination of a and b
will just end up on this line right here, if I draw
it in standard form. It'll be a vector with the same
slope as either a or b, or same inclination, whatever
you want to call it. I could never-- there's no
combination of a and b that I could represent this vector,
that I could represent vector c. I just can't do it. I can add in standard form. I could just keep adding scale
up a, scale up b, put them heads to tails, I'll just get
the stuff on this line. I'll never get to this. So in this case, the span--
and I want to be clear. This is for this particular a
and b, not for the a and b-- for this blue a and this yellow
b, the span here is just this line. It's just this line. It's not all of R2. So this isn't just some kind of
statement when I first did it with that example. It's like, OK, can
any two vectors represent anything in R2? Well, no. I just showed you two vectors
that can't represent that. What is the span of
the 0 vector? I'll put a cap over it, the 0
vector, make it really bold. Well, the 0 vector is just 0,
0, so I don't care what multiple I put on it. The span of it is all of the
linear combinations of this, so essentially, I could put
arbitrary real numbers here, but I'm just going to end
up with a 0, 0 vector. So the span of the 0 vector
is just the 0 vector. The only vector I can get with
a linear combination of this, the 0 vector by itself, is
just the 0 vector itself. Likewise, if I take the span of
just, you know, let's say I go back to this example
right here. My a vector was right
like that. Let me draw it in
a better color. My a vector looked like that. If I were to ask just what the
span of a is, it's all the vectors you can get by
creating a linear combination of just a. So it's really just scaling. You can't even talk about
combinations, really. So it's just c times a,
all of those vectors. And we saw in the video where
I parametrized or showed a parametric representation of a
line, that this, the span of just this vector a, is the line
that's formed when you just scale a up and down. So span of a is just a line. You have to have two vectors,
and they can't be collinear, in order span all of R2. And I haven't proven that to you
yet, but we saw with this example, if you pick this a and
this b, you can represent all of R2 with just
these two vectors. Now, the two vectors that you're
most familiar with to that span R2 are, if you take
a little physics class, you have your i and j
unit vectors. And in our notation, i, the unit
vector i that you learned in physics class, would
be the vector 1, 0. So this is i, that's the vector
i, and then the vector j is the unit vector 0, 1. This is what you learned
in physics class. Let me do it in a
different color. This is j. j is that. And you learned that they're
orthogonal, and we're going to talk a lot more about what
orthogonality means, but in our traditional sense that we
learned in high school, it means that they're 90 degrees. But you can clearly represent
any angle, or any vector, in R2, by these two vectors. And the fact that they're
orthogonal makes them extra nice, and that's why these
form-- and I'm going to throw out a word here that I
haven't defined yet. These form the basis. These form a basis for R2. In fact, you can represent
anything in R2 by these two vectors. line. I'm not going to even define
what basis is. That's going to be
a future video. But let me just write the formal
math-y definition of span, just so you're
satisfied. So if I were to write the span
of a set of vectors, v1, v2, all the way to vn, that just
means the set of all of the vectors, where I have c1 times
v1 plus c2 times v2 all the way to cn-- let me scroll over--
all the way to cn vn. So this is a set of vectors
because I can pick my ci's to be any member of the real
numbers, and that's true for i-- so I should write for i to
be anywhere between 1 and n. All I'm saying is that look, I
can multiply each of these vectors by any value, any
arbitrary value, real value, and then I can add them up. And now the set of all of the
combinations, scaled-up combinations I can get, that's
the span of these vectors. You can kind of view it as the
space of all of the vectors that can be represented by a
combination of these vectors right there. And so the word span,
I think it does have an intuitive sense. I mean, if I say that, you know,
in my first example, I showed you those two vectors
span, or a and b spans R2. I wrote it right here. That tells me that any vector in
R2 can be represented by a linear combination of a and b. And actually, just in case
that visual kind of pseudo-proof doesn't do you
justice, let me prove it to you algebraically. I'm telling you that I can
take-- let's say I want to represent, you know, I have
some-- let me rewrite my a's and b's again. So this was my vector a. It was 1, 2, and b was 0, 3. Let me remember that. So my vector a is 1, 2, and
my vector b was 0, 3. Now my claim was that I can
represent any point. Let's say I want to represent
some arbitrary point x in R2, so its coordinates
are x1 and x2. I need to be able to prove to
you that I can get to any x1 and any x2 with some combination
of these guys. So let's say that my
combination, I say c1 times a plus c2 times b has to be
equal to my vector x. Let me show you that I can
always find a c1 or c2 given that you give me some x's. So let's just write this right
here with the actual vectors being represented in their
kind of column form. So we have c1 times this vector
plus c2 times the b vector 0, 3 should be able to
be equal to my x vector, should be able to be equal to my
x1 and x2, where these are just arbitrary. So let's see if I can
set that to be true. So if this is true, then the
following must be true. c1 times 1 plus 0 times c2
must be equal to x1. We just get that from our
definition of multiplying vectors times scalars
and adding vectors. And then we also know that
2 times c2-- sorry. c1 times 2 plus c2 times 3, 3c2,
should be equal to x2. Now, if I can show you that I
can always find c1's and c2's given any x1's and x2's, then
I've proven that I can get to any point in R2 using just
these two vectors. So let me see if
I can do that. So this is just a system
of two unknowns. This is just 0. We can ignore it. So let's multiply this equation
up here by minus 2 and put it here. So we get minus 2, c1--
I'm just multiplying this times minus 2. We get a 0 here, plus 0
is equal to minus 2x1. And then you add these two. You get 3c2, right? These cancel out. You get 3-- let me write it
in a different color. You get 3c2 is equal
to x2 minus 2x1. Or divide both sides by 3,
you get c2 is equal to 1/3 x2 minus x1. Now we'd have to go substitute
back in for c1. But we have this first equation
right here, that c1, this first equation that says
c1 plus 0 is equal to x1, so c1 is equal to x1. So that one just
gets us there. So c1 is equal to x1. So you give me any point in R2--
these are just two real numbers-- and I can just perform
this operation, and I'll tell you what weights to
apply to a and b to get to that point. If you say, OK, what combination
of a and b can get me to the point-- let's say I
want to get to the point-- let me go back up here. Oh, it's way up there. Let's say I'm looking to
get to the point 2, 2. So x1 is 2. Let me write it down here. Say I'm trying to get to the
point the vector 2, 2. What combinations of a
and b can be there? Well, I know that c1 is equal
to x1, so that's equal to 2, and c2 is equal to 1/3
times 2 minus 2. So 2 minus 2 is 0, so
c2 is equal to 0. So if I want to just get to
the point 2, 2, I just multiply-- oh, I
just realized. This was looking suspicious. I made a slight error here,
and this was good that I actually tried it out
with real numbers. Over here, when I had 3c2 is
equal to x2 minus 2x1, I got rid of this 2 over here. There's a 2 over here. I divide both sides by 3. I get 1/3 times x2 minus 2x1. And that's why I was like, wait,
this is looking strange. So I had to take a
moment of pause. So let's go to my corrected
definition of c2. C2 is equal to 1/3 times x2. So 2 minus 2 times x1,
so minus 2 times 2. So it's equal to 1/3 times 2
minus 4, which is equal to minus 2, so it's equal
to minus 2/3. So if I multiply 2 times my
vector a minus 2/3 times my vector b, I will get
to the vector 2, 2. And you can verify
it for yourself. 2 times my vector a 1, 2, minus
2/3 times my vector b 0, 3, should equal 2, 2.