Welcome to our lesson on Linear Combinations
and the Span of Vectors.
Given vectors v sub one through v sub k are in Rn,
and c sub one through c sub k are scalars,
vector X is a linear combination
of the vectors v sub one through v sub k
if the sum of the products of the scalars and the vectors
is equal to vector X.
As an example, all vectors in R2
are linear combinations of the unit vectors i and j
or the vectors one, zero and zero, one.
These are also called the standard basis vectors in R2.
As an example, the vector three, negative two
is a linear combination of the vectors one, zero
and zero, one since three times the vector one, zero
minus two times the vector zero, one
is equal to the vector three, negative two.
We can easily check this.
Notice three times one minus two times zero is three
and three times zero, minus two times one is negative two.
So in this example,
the scalars c sub one and c sub two
are equal to three and negative two,
and the vectors v sub one and v sub two,
are the unit vectors i and j,
and the vector three, negative two
is a linear combination of the unit vectors i and j.
The vectors v sub one through v sub k
don't have to be the standard basis vectors.
For example, the vector eight, negative five
is a linear combination of the vectors three, 0.5
and one, negative three,
since two times the vector three, 0.5
plus two times the vector one, negative three
is equal to the vector eight, negative five.
Checking, notice two times three
plus two times one is eight.
For an example, in R3,
the vector negative four, negative 13, five
is a linear combination of the vectors four, three, five,
negative two, negative five, three and three, five,
negative two since negative one
times the vector four, three, five
plus six times the vector negative two,
negative five, three,
plus four times the vector three, five, negative two
is equal to the vector negative four, negative 13, five.
Not all vectors can be written as linear combinations
of any set of vectors.
As an example, the vector five, negative eight
cannot be written as a linear combination
of the vectors two, four and negative three, negative six.
Let's take a closer look at why this is.
If we could write the vector eight, negative five
as a linear combination of the two vectors,
there must exist a c sub one and a c sub two,
such that c sub one times the vector two, four,
plus c sub two times the vector negative three, negative six
equals the vector eight, negative five.
And now if we perform the scalar application
on the left and then add the two matrices,
we would have a column matrix where the first entry,
we would have two by one column matrix
where the first entry is two c sub one
minus three c sub two,
and the second entry would be four c sub one
minus six c sub two.
Which is still equal to the vector on the right,
the vector eight, negative five.
From here, we can write a system of equations
where the first equation is two c sub one
minus three c sub two equals eight,
and the second equation is four c sub one
minus six c sub two equals negative five.
To solve the system,
we write an augmented matrix
and then write the augmented matrix
in reduced row-echelon form.
Looking at the matrix in reduced row-echelon form,
notice in the second row, we have zero, zero, one
which indicates that zero equals one,
which is never true since the last row of the matrix
in reduced row-echelon form is false,
this indicates the system has no solution,
which indicates there are no values
of c sub one and c sub two
where the vector eight, negative five
is a linear combination of the vectors two, four,
and negative three, negative six.
And now let's talk about the span of a set of vectors.
The span of a set of vectors
is a set of all possible linear combinations of the vectors.
So the span of the vectors v sub one through v sub k
contain all the vectors in the form of the sum
of the scalars and the vectors v sub one
through v sub k.
And since we know every vector in R2
can be written as a linear combination
of the unit vectors i and j
or the vectors one, zero, and zero, one,
the span of the unit vectors i and j is R2.
Similarly in R3, the span of the vectors one, zero, zero,
zero, one, zero and zero, zero, one is R3,
because any vector in R3
can be written as a linear combination
of these three vectors.
And now consider the span of the vector two, three.
Because we have a single vector here,
the linear combinations of this vector
would just be scalar multiples of the vector.
So let's take a look at this graphically.
Here we have the graph of the vector two, three,
and now let's take a look at the linear combinations
of this vector, which again,
will just be scalar multiples of this vector
because we have a single vector.
So here we see many of the linear combinations
of the vector.
Notice that all the vectors fall on the same line,
which indicates the span of the vector two, three
is a line containing all of these vectors,
which would be this yellow line.
And now it's a term in the span of the vectors,
one and negative one and three, one.
And again, let's look at this graphically.
The vector one, negative one is graphed in green,
the vector three, one is graphed in red.
And now we'll look at some of the linear combinations
of these two vectors,
which will be the sum of scalar multiples
of these two vectors.
And therefore the blue vector
will always be a linear combination
of the two given vectors.
Notice how we're tracing out points
in every quadrant of the coordinate plane.
In fact, we could trace out all points
on the coordinate plane,
by determining linear combinations of the two given vectors,
which means a span of the two given vectors
is the entire coordinate plane or R2.
And now let's also consider the span of two vectors in R3,
the vectors one, one, one and one, two, three.
It's a little harder to see here,
but the two given vectors are the green and red vectors
and the blue vector is linear combination
of the two given vectors.
And [indistinct] other linear combinations
of the two given vectors.
I can remember the blue vector
is a linear combination of the two given vectors.
So what we'll notice here
is a linear combination of the two given vectors
is always in this yellow plane
and the linear combination could trace out
all the points in the yellow plane and therefore the span
is the yellow plane.
Let's go ahead and summarize our findings.
The span of the vector, two, three is a line,
the span of the vectors one and negative one and three, one
is the coordinate planar R2,
which would be the span for any two vectors in R2,
as long as the vectors are not scalar multiples,
and neither is a zero vector.
The span of the vector is one, one, one and one, two, three
is a plane, and this will also always be true
for two vectors in R3
as long as the vectors are not scalar multiples
and neither it is a zero vector.
And then to finish the lesson,
we will need to determine if a given vector
is a linear combination of a given set of vectors
to know if a vector is in the span of a set of vectors.
We will look at examples of this in the next several videos.
I hope you found this helpful.