Hi everyone. This is Professor Yong-Jin Yoon from from KAIST. This is 3rd session of the week second for the beginner math for the AI beginners Part 1, which is about linear algebra. For the last session you just taste of what is the linear algebraic equations. Actually many problems in the nature and also in engineering and physical sciences are formulated in terms of the same system of linear algebraic equation. I give some example. This is the temperature of your eyeball from the cornea to the back side of the eye. As you expected, the cornea, which is contacted with the air, should be the lowest temperature, and inside is most highest temperature because it's inside of our body. The temperature profile in the human eye was obtained from the boundary elements method by solving a system of 470 linear algebraic equations in 470 unknowns. This is definitely very difficult to solve by writing by hand. You should make a model to solve this system of linear algebraic equation because too many unknowns here and two different boundary conditions for that. When you use this biomedical application, this system of linear algebraic equation is also be used. This is second example. Now, this is actually my research topic. We use a lot of system of linear algebraic equation when we simulate the motion of the hearing. This is from the eardrum and vibration of the eardrum stimulate the cochlea, and then those fluid inside the cochlea is vibrated by the motion of the eardrum vibration. This very complicated motion also can be solved in system of linear algebraic equation. This is another simple model for the computation of the human ear. Also, we use the computer simulation for simulating the motion of the basilar membrane, also cochlea mechanics. Definitely, followed by the course name, the deep learning and machine learning also can be used by using this linear algebra is very essential part to build up this deep learning algorithm, such as deep neural network which we studied. When you calculate this matrix, 3 by 2, 2 by 4, 4 by 2 matrix, we need to know that basic linear algebraic equation to find out this black box, which is deep neural network; the AI engines. So that this is the power of the linear algebra and this is the why we need to learn the linear algebra to understand and to develop more about artificial intelligence like engine such as deep learning. Up to here, I just introduced the why we need to know about linear algebraic equation. Here is the system of N linear algebraic equation in N unknown. For example, I just introduced 470 unknowns, linear algebraic equation, but here, N unknowns. In general, we can express the system of N linear algebraic equation like this; a_11 times x_1 plus a_12 times x_2 plus a_1N times x_1 equals b_1. This is first linear algebraic equation, and 2nd linear algebraic equation is a_21 x_1 plus a_22 x_2 plus a_2N x_N equal b_2. Keep going like this, and last linear algebraic equation is a_N1 x_1 plus a_N2 x_2 plus a_NN x_N equals b_N. We can say N linear algebraic equation in N unknowns. Here the constant a_ij is the coefficient of the unknowns x_j in the i-th equation, and b_i is the constant term in the i-th equation. This complicated system can be written in a very simple form with a matrix form AX equals B. This A is system of linear algebraic equation like some constant, a_ij, x is unknown and b is the constant in the right-hand side. So a, we can formulate matrix a as with the constant of the linear algebraic equation, multiplication to the each unknowns x_1 to x_N. This is a N by N matrix a_11 to the a_1N, and a_N1 - the a_NN. X is unknown, so we can say this is N by 1 matrix, or the Nth dimensional vector. B equals b_1, b_2 to the b_n. For example, in a simple case, we can bring out the example for the two unknowns, x and y and two equations like this. The first linear algebraic equation is 2x plus 3y equals 10, and second is minus x plus y equals 0. This one is also an example system of linear algebraic equation. We can formulate this equation with the matrix form like just take the matrix A element with constant which is multiplied with unknowns x and y 2, 3, minus 1, 1 times, and then x are unknowns here in x and y because the B is what, 10, 0 like this. We can formulate the matrix form from the system of linear algebraic equation. These 2,3 minus 1, 1 times x, y we can express with different way, 2x plus 3y with minus x plus y, like this. Then, if we match to the left-hand side, to the right-hand side matrix together, then you can achieve the original system of linear algebraic equation 2, which is 2x plus 3y equals 10 and minus x plus y equals 0. Let me talk about a solution of linear algebraic equation. Let's consider a single linear algebraic equation, which is c_1x_1 plus c_2x_2 plus c_N x_N equals d_N. Here, x_1, x_2 to the x_N is the unknowns for this single linear algebraic equation. If this x_1 to x_N equals alpha_1 to alpha_N is said to be a solution of our linear algebraic equation if the left-hand side of the equation equals this right-hand side, when we replace x_1, x_2 to x_N by alpha_1, alpha_2 to alpha_N respectively. If you just plug in this x_1 to the x_N with the alpha_1 to alpha_N to the equation if the regional to the calculation of left-hand side c_1 alpha_1 plus c_2 alpha_2 plus c_N alpha_N equals d_N. Then what? Alpha_1 to alpha_N is the solution of this single linear algebraic equation. For example, let me introduce the x plus 2y minus z equals 0. How many unknowns here? Three unknowns, x, y, z. Let's say we bring out x, y, z with minus 1, 2, 3. If we plug in these, x equals minus 1, y equals 2, z equals 3 to the linear algebraic equation, then it becomes 0. Left-hand side is 0, right-hand side 0, so minus 1, 2, 3 is the solution of this linear algebraic equation. However, if we use x, y, z, which is 2 minus 1, 3, then the value of the left-hand side is different from the right-hand side 0. So 2 minus 1, 3 is not the solution for this linear algebraic equation. We can find many other solution here easily. Solution of linear algebraic equation is that we can say if x_1, x_2, x_N equals alpha_1 to alpha_N, your solution of each and every linear algebra equation in the system, then it is said to be a solution of the system. It is possible that our system of linear algebraic equation has no solution. For example, x plus y equals 10, x plus y equals 5. This kind of a system is said to be inconsistent. Because sometimes the x plus y become 10, sometime the x plus y equals 5. This is inconsistent. In this case, this system of linear algebra equation has no solution. In other case, it is possible that the consistent system has more than one solution. For example, x plus y equals 10,2x plus 2y equals 20. Actually, second linear algebraic equation is same as for linear algebraic equation x plus y equals 10. The system really contains only one linear algebraic equation with two unknowns. We can find infinitely many solutions for the system. Up to here, we study about the characteristics of the linear algebraic equation with afforded whether there is a consistent system or incontinence system where many solution, where unique solution. For the next session, we will study more details about this consistency of the system of linear algebraic equation and the unique solution and infinitely many solution case. Thank you very much.
