The lecture begin with Chapter 1, systems of linear equations. In this chapter, we're forced to develop the basic theory of linear systems of first-order differential equations, and then introduce the method of self solving, such systems with constant coefficients. Let's consider the following system. The force to the system in its normal form. Which means the x1 prime of t is equal to a sub 1, 1 times x sub 1 plus a sub 1, 2, t times x of 2 plus and the a sub 1 and t of the x sub n and plus f_1 of t. That's the first equation, and by the same pattern. The second one is x2 prime is equal to a sub 2, 1 of t of x1 plus a sub 2, 2 of t of x2 plus, and so on a sub 2 sub n, a sub 2 n of t, x sub n plus f of 2 of t, and finally we have x of n of prime of t is equal to a sub n 1 of t, x1 plus a sub n 2 of t of x2 and plus a sub n, n of t of xn and plus f sub n of t. We have totally unknown functions from x sub 1 of t to x sub n of t, and they satisfy the first-order, linear system of equations. Where we are going to assume that in this expression all the coefficients, a sub Ij of t and those the fj of t. All those functions are continuous on some common interval I. If all those the right-hand side, if fg of t are identically equal to 0 and this interval I, then call the system. Let's call the system. This is a system of differential equation 1, then call the system 1 to be homogeneous. Otherwise it means what? At least one of those, the f_1 and f_2 in the episode where at least one of them is not identically 0 on I. Otherwise, call it to be non-homogeneous. This notion of the homogeneous or non-homogeneous it is the same concept we have introduced in differential equation, the Part 1 on the basic theory for single linear equations. Let's first introduce some simple notation to express the system of such equation. Let's introduce the following notation. Let the X of t this is a column vector having the components from x1 of t, x2 of t through the x1 of t, this is the column vector, unknown column vector. Capitally A of t. This is the environment matrix with entries A small a_i j of t. Where the both indices I and j, they are moving from one to the n so that this is the n by n matrix. The coefficient matrix, this will be called as a coefficient matrix. Finally, let the capital F of t to be the column vector having component the F1 of t and the f of 2 of k through the f of n of t. Using this matrix notation or the vector notation, we can express, one, there's a system, one, as the capital X prime of t is equal t, capital A of t times x of t, and the plus capital F of t. This is a short notation for matrix notation for the given linear system of differential equations. Now the Luxor reminded the concept we have introduced a couple of minutes ago. This is called the homogeneous. This differential equation is called homogeneous safe. This vector F, capital F is equal to 0 vector way. All the components are 0. Non-homogeneous otherwise, if not. The basic concept related to the linear force to the system. Next, consider the very simple example, where we have three unknowns. Instead of the X1, X2, X3, I will use the x, y, t. Let's just see that in the following situation. Example 1, consider the system of linear equations. X prime of t is equal t, 3x minus y plus 2z minus t, and y prime of t. That is equal to minus 2x plus 2y plus 7t. The z prime of t, that is equal to 2x plus 7y minus 5z minus 3 of t. Let's consider the following, the first-order linear system of differential equations where we have three unknowns, say x_t and y of t and z of t. Represented in the matrix form, let's introduce capital X is equal to unknown vector X, Y, Z. Then lastly read out the coefficient matrix capital A, that you can read it from here. From the very first one you read the 3 minus 1 and 2. From the second line, minus 2 and 2. There is no z, so that then means 0. From the last line, you have 2, 7 minus 5. This is the coefficient matrix A. Finally, the non-homogeneous tone capital F is equal 2. From the force T we have the minus t from the second year of 7t, from the last year of a 3t. Using this matrix notation, we can see that we can rely to this system of equation into the capital X prime is equal to A times the capital X plus capital F. This is just notation for this given system of differential equation. The expressing the system of equation given individually into the matrix form is quite a straightforward task.