[MUSIC] Hi everyone, this is Professor Yang Jian Yang from Kaist. Now we are in the fourth session of the second year for Beijing mess for the beginners of AI Part one in algebra. So up to last session, we study about the system of linear algebraic equation, and whether the study of our case, whether there is no solution and many solutions, right. Record if there is a system of linear algebra equation has no solution, it means inconsistent system, for the consistent system, we can divide you into the two different case. First cases for the infinitely many solutions or the new solution, so if a consistent system of linear algebraic equation has only one solution, we say the system has a unique solution to summarize a system of linear. Algebraic equation can either be consistent or inconsistent, if the system is consistent, the system can have either a unique solution, or if you only many solutions. So given a system of linear Algebraic equation such as AX = B, how do we know that whether it is consistent or not, right? By looking at the matrix form A, can you differentiate those AX = B as a solution or consistent solution or not, or if it is consistent, how do you find all solutions? So this is the major part of the linear algebraic equation to serve that this system of linear algebraic equation, the one of the best ways to find the solution of the linear algebraic equation. Instead reducing the system AX= B to a simpler form but equivalent system UX=C, so for the system of miniaturization, A X =B, and UX=C are equivalent if they have exactly the same solution. So if we can work out the solution of UX=C, then if you find solution X here then we would have solved AX=B too. So the region is why because their solution is saying so it means AX=B, which is a system that we are the application AX=B, is equivalent to the UX=C. If the scare matrix U is an upper triangular matrix, the system UX=C would be simpler. You know for us to work out a solution, so let me introduce what is the upper triangular matrix? So this is the example of upper triangular matrix. So as you see, the upper triangular matrix has only element, upper triangular part from the diagonal element. Below the diagonal element here 1, 1, 2, 3, the upper triangular matrix element is our general zero. So again to solve AX=B, we can reduce it to an equivalent system, UX=C, where the U is triangular matrix. How can this be done? Let's write A X=B in a tableau form, A/B like this. So example, there is a let's say this is a three unknown and system of linear algebraic equation, we have three linear algebraic equation here. So 2x+3y+4z=6, and second equation is 3x+5y-2z=7, and x+10y+5z=9, and we can simply write this occasion to the tableau form, just using the constant here. 2 3 4, 3 5 -2, 1 10 5, which is what, the elements of the matrix A and 6 7 9 is what, element of the right hand side B, matrix B. So when you use register mate cooperation in a systematic manner to reduce this tableau form to become U/C. So there are two types of legitimate row operations, so first row is what? First row of operation is what? Ri can be changed interchanged with Rj, so you can interchange i and j slow each other. The second rule is you can modify certain row, for example i's row Ri with multiplication of alpha Ri plus beta Rj. So we can use row of j to change row of i to become our alpha Ri plus beta Rj, here important is that the constant alpha cannot be zero, why? If you put alpha zero and let's say beta is one here, then let's say for the first row, R1 one become R3, which means just disappear. Ri, R1 is disappear, R1 means that well first linearity algebraic equation become third linear algebraic equation. So you cannot put alpha as a zero here, so simple rule to observe here is that, in changing a tableau by second type of row operation, keep one row fixed and use the fixed row to change other row(s). This is the two types of register mate row operations, you should remember that, but up to here is a bit difficult to understand, so don't worry about it. For the next week we are going to study about this legitimate cooperation with many example to solve this system of linear algebraic equation. Thank you very much.
