I'm going to give you a quick introduction to linear algebra. This will not be a complete course. It's just a very small introduction enough to give you an understanding so that you can explore tools like cuBLAS the BLAS, and other CUDA linear algebra libraries. I would encourage you to look for resources like other Coursera courses or YouTube videos, or even textbooks like Gilbert Strang's linear algebra book. Let's go over some very basic concepts. Consider the formulation of the description of a vector or a line, as it's stated here, with a_1x_1 to a_nx_n equals a constant b, where x_1 to x_n are variables and a_1 to a_n are scalar values. Often, what you'll want to do is you're given a series of these equations and you want to solve for each of those x or our variables. A common way to do that is to construct a matrix where you take all of the variables in the scalar multipliers and put them on the left, order them such as x, then y, then z. On the right side, you put what that constant that that equals. The number of rows in the matrix is the number of equations, the number of columns, number of variables, plus a constant. Each row as I said before is an equation, which means it describes a line or a vector, and the collection of those values is a matrix in multiple dimensions. Two common matrix operations is multiplying a matrix by a scalar or a constant value. In this case, you're multiplying the matrix by two, so all the values in that matrix get multiplied, and correspondingly, the result gets placed into the same size matrix. Dot product is the other type of multiplication. You take two matrices and you multiply them. How we describe the size of matrices is rows and columns. In our example here, we have a two by three matrix being dot product by a three by two. You end up with a two-by-two matrix. On dot product, multiplication works for matrices is you go by the rows in the left matrix and you multiply them by the right matrix, the columns. In the case of the first row, in the second position and the resultant matrix, it is the first row times by the second column. It's 1 times 8, plus 2 times 10, plus 3 times 12, which is 64, and you proceed through. You end up with a square matrix, which is often a good place to be because you could perform certain operations only if you have a square matrix. One of the most common operations to be performed on single matrices is that you calculate a determinant. This can only be done on square matrices. We show a two-by-two matrix here, but in reality, it can be any square matrix, three-by-three, four-by-four. It does get more complicated, and there are ways to calculate it, but this is one of the reasons why people like computers to calculate these values because it can be really complex. What would you use something like a determinant for? You can determine if three lines are colinear, which means that even though the scalars are different, they actually overlap. Just remember basic geometry is that any line can be represented by different equations, but actually have the same trajectory, start at the same place and end in the same place. Determining the size of a triangle, you can use determinant. If the determinant equals zero, that means that at least two of the lines are colinear. If it's greater than or less than zero, that means that all three lines are not co-linear. They start and end at different places, they overlap, they intersect, and therefore, in multiple dimensions, you have a triangle and therefore, you know the size of it then by the determinant.
