 Welcome back to our lecture series linear algebra done openly. As usual, I'm your professor today, Dr. Andrew Misseldine. This is the first video for section 2.3 about linear independence, which is a pretty big important notion for linear algebra. We don't usually like to put the word linear in front of something, unless it's a big deal in linear algebra. In this video, we're going to find what linear independence means, and in the subsequent videos, I'll just give you some examples on how one can determine whether a set is linearly independent or not. So a set of vectors, let's say a1, a2, all the way up to an, these are going to be inside the vector space fn. We say a set of vectors is linearly independent if when we look at the vector equation x1 a1 plus x2 a2 all the way up to xn an, that equals zero only when we have the trivial solution. What do you mean by the trivial solution here? Well, when you have a vector equation that's equal to zero, right? So we have a linear combination equal to zero. I want you to be aware that we could rewrite this as a matrix equation ax equals zero. We could also write this as a linear system. That linear system would be the augmented matrix a augment zero. This is an example of the homogeneous, this is a homogeneous system, all right. And when one works the homogeneous system, a homogeneous system is always consistent because you have this so-called trivial solution where you take the vector x and you set it equal to the zero vector. Notice that if I set each of these coefficients to zero, zero, zero, zero, you're going to get zero times a1. Well, I have no idea what a1 is necessarily. It's just a vector, but zero times any vector is going to give you zero. So you're just going to get the zero vector over and over and over again. Well, if you add the zero vector, zero vector together in times, you're still going to get the zero vector. So the solution x equals zero is always a solution to the homogeneous system. And we call this the trivial solution because it's always possible. But on the other hand, are there other solutions to this homogeneous system? That's really the question at hand. Now, we say that the vectors a1 through an are linearly independent if there are no other solutions. And otherwise, we say the set of vectors is linearly dependent, which would mean that there's a non-trivial solution to this homogeneous system. So imagine the system was, that is, if the homogeneous system does have a non-trivial solution, in that situation, those vectors would be linearly dependent. And so we would then say that such an example, so we have some specific coefficients c1, c2, up to cn, we have specific numbers, which when combined together, this linear combination gives us zero. But we also have that not all of the coefficients are zero, at least one of them is not zero. This is an example of a linear dependence relationship, or what we'll just call a dependence relationship for short, that there's some way of combining the vectors together in a non-trivial way to give you zero. And as I mentioned earlier, we can visualize this vector equation as a matrix equation, right? If a is the matrix whose column vectors are a1, a2, up to an, then the column vectors of the matrix are linearly independent, if and only if the matrix equation ax equals zero has no non-trivial solutions. And it's like we saw before, if you take a matrix whose column vectors give it here, you take a vector x1, x2, up to xn, the matrix product gives you this linear combination. So the two are related to each other, and a set of vectors is linearly dependent, if and only if the equation ax equals zero has a non-trivial solution.