Hey everyone, in this video I want to do this linear algebra problem.
In this problem we're given a basis B for a vector space, which we're given is R3. It's a basis for
any three-dimensional vector space. We're given a vector lowercase v which is (0, 1, 3). It's just
an example three dimensional vector. And we're asked to find the coordinate vector v of B. So v
of B is also known as the coordinate vector. Let's get into how we solve this problem and what all
these different words that I just said actually mean in linear algebra. We're given in the problem
we have a basis, which is a set of vectors, as we can see we have a set of three vectors
and each vector is three-dimensional. And we are given that we have a basis. It's a basis for R3.
We're given a vector lowercase v and we have to find the coordinate vector. Let's define a lot of
these terms and the definitions are going to help us be able to solve for the coordinate vector.
By definition, a basis is just a set of vectors and by definition, for any vector in a vector
space--in this case, any v in R3--there is a unique linear combination of the basis vectors
that equals any vector v. That's the definition of basis. If we have three basis vectors that
form our basis--I'm going to label those b1, b2, b3. These are the basis vectors that form our
basis--then for any vector v in the vector space, which in this case our vector space is R3, we can
write out a mathematical equation for the vector v as being equal to a linear combination of b1,
b2, b3. Let's say--I'm just going to write out the coefficients using a. Let's say a1 times b1
plus a2 times b2 plus a3 times b3. There has to be some linear combination of the basis vectors
that will give you any vector in the vector space. A vector space is usually written in
capital V. In this case we're given that this is just any three-dimensional vector. And a vector
space is just a set of vectors itself. Any v in a set of vectors known as the vector space,
this is the definition that we're working with. A coordinate vector v of B is sometimes known as the
B--capital B--representation of v. We're given a vector, we're given the basis for a vector space,
and basically we want to be able to transform the basis to this vector. And the way we do
that is with the coefficients a1, a2, a3. The coordinate vector v of B is equal to a vector of
these coefficients. In this case, v of B is equal to a1, a2, a3, and we have to solve and find a1,
a2, a3 to solve this problem. Now that we know what we have to solve for, we know because we're
given in the problem, we're given lowercase v, so we're given the vector that we want to transform
from a basis to the vector v. We're given the vector v is equal to (0, 1, 3) and we're given the
basis. We're given the basis vectors b1, b2, b3. We're given v. We have enough information in this
problem in order to find the coordinate vector. We just have to plug in v and the three b's to
solve for a1, a2, a3. Let's go ahead and do that next in this problem. I'm just going to go ahead
and plug in everything that I just mentioned. V is equal to (0, 1, 3). That's going to be equal to a1
times b1, which is (1, 1, 0) plus a2 times b2, which is (0, 1, 1) plus a3 times b3, which is (1,
0, 1). Next I'm just going to scalar multiply the coefficients to the vectors. That's going to be
equal to a1, a1, 0 plus 0, a2, a2 plus a3, 0, a3. And then to find the values of a1, a2, a3, we can
solve this as a system of linear equations. We can say that a1 plus 0 plus a3 has to be equal to 0.
We know that a1 plus a2 plus 0 has to be equal to 1. And we know that 0 plus a2 plus a3 has to be
equal to 3. This is a system of linear equations with three equations and three unknowns. We can
find a unique solution for a1, a2, a3. Using the first equation, I'm going to say that that
means that a1 is equal to -a3. I'm going to plug that in to the second equation, so a1 is equal to
-a3 plus a2 is equal to 1. That means that a2 is equal to 1 plus a3. I'm going to plug that into
the third equation. Plugging in for a2, that's going to be 1 plus a3 plus a3 is equal to 3.
That's going to give us that 2 a3 is equal to 2, which means that a3 is equal to 1. We've solved
for a3. I'm going to go back in and plug that into a2. One plus 1 is equal to 2. So a2 is equal to 2.
Then that means that a1 is equal to -1. And now we've solved for a1, a2, and a3. I'm just going
to plug that in to the vector form and we have the solution for this problem. The coordinate vector
v of B is going to be equal to the vector (-1, 2, 1) and that's the solution for this problem.
*chiptune music*