Hi everyone. This is Professor Yongjin Yoon from KAIST. This is second unit in a first session for the mathematics for AI beginner part 1, which is about linear algebra. From last session and last week, we started about what is AI, and why we need to learn the mathematics. Paging mathematics such as linear algebra and vector calculus to understand AI more intensively. Let me start linear algebra. We start with a review of basic matrix. Maybe you may learn from your middle school and high school. Then after that, we are going to study about the system of linear algebraic equation, which is AX equals B. Here, A and X and B is a vector and matrix, and by using the findings solution, by using the row operation. After that, we are going to study about the inverse of a square matrix A, also by using the row operation. Then later, we will study about determinant of a square matrix A by calculating the determinant also by using row operations. What you are going to learn during this course is about this. In big picture, we are going to study about how those three major contents are related to and connected to each other for the AI application. In-between, we look over the vector of the space and also linearly independent vectors, such as for the Matrix Eigenvalue Problem, for Diagonalization Problem, AI Application for such as Deep Learning and Support Vector Machine. Let me leave the paging matrix, so what is a matrix? Matrix, you may think about like the movie like Matrix is similar meaning. Matrix is kind of a system which describe some physical world. That's why the movie, Matrix take their name like matrix here. In terms of mathematical terminology, matrix is an M by N matrix in general. M by N matrix is an array of M by N numbers enclosed with a pair of a bracket and arranged in M rows and N column. For example, for these 3 with bracket. Without bracket 3 is what? Just number. But with bracket, it means one-by-one matrix. One-by-one matrix we call scalar. The second one, 2, 1, minus 1, 2, and bracket, this is two-by-two matrix. Third one, is 1, pi, square root of 2, 1/2, 5, 6 with bracket. This is three-by-two matrix, three rows and two column. That's why, we call this is three-by-two matrix. The last one, 3, 6, 9, 2, 3, 9, bracket, this is what? Two-by-three matrix, number of row is two, number of a column is three, so this is two-by-three matrix. Matrix is just put the numbers in the bracket. The numbers in the matrix, we called this one as element. Element for example, in 2, 1, minus 1, 2, an element is what? Two, 1, minus 1, 2, there are four elements. We called those numbers which formulate the matrix, we call it element. Also, this is can be considered as a bit data in the AI. There is some special shape of the matrix, we called square matrix. Square matrix, definition is the number of rows equals the number of a column. For example, the second example, 2, 1, minus 1, 2. Is two-by-two matrix, so we called this two-by-two matrix as square matrix. There will be three-by-three matrix. Also here 3, is also a matrix with element only of 3, one-by-one, is also square matrix. But there will be 3 by 3 matrix, 4 by 4 matrix, M by N matrix right there. When you describe matrix, we normally use bold capital letters, such as A, B, P, Q, like this. For example, we can write matrix A as capital letter A, like this. For comparison, when you use lower letter then it represent normally vectors. Let me talk about the notation. How we describe a matrix A with M by N matrix. We normally denote an element in the ith row and jth column of A by a_ij. This A means M by N matrix, and the a_ij means elements, the numbers inside of the matrix A, element in the ith row and jth column. We can simply describe M by N matrix. How many elements there? M times N. Can be a lot of elements, but in a very simple notation, mathematically, we can write the matrix A equals bracket with element a_ij. This can be expanded to the more details notation like this. A_11 is element in first row and first column, a_12 is element in the first row and the second column, a_1N is element in the first row, Nth column. Like this, a_11, to the a_1N, and a_21 to the a_2N, keep going, a_M1 to the a_MN. This is M by N matrix. But we can easily denote like this. Let matrix A equals like this, a_ij, and matrix B equals b_ij be the both M by N matrix, which is same order. This M by N, we record this one as our order of the matrix. Let me define equality of the matrix. Matrix A and B are said to be equal. That is, our matrix A equals B if the element a_ij equals element of b_ij is same. This is mathematical definition of equality of matrix. For example, there's to left-hand side, let's say this is matrix A, right-hand side matrix, let's say this matrix B. To make the equality of matrix A and B, what is the condition? Like this, A plus B equals C. A plus B is element of A in the first row and first column. C equals element of matrix B in the first row and first column. A plus b equals c, and c equals 3, and c equals d, and a minus b equals b. This is equality of matrix. Let me introduce definition of addition of matrix. If matrix A plus matrix B equals C then matrix C equals c_ij is M by N and the element of matrix C, which is c_ij equals a_ij plus b_ij. Simply, the addition of matrix, just the addition of element in the right position, same position in ith row and jth column. For example, let's say matrix A is 1, 2, 3, 4, 5, 6 plus matrix B is 7, 8, 9, 10, 11, 12. Then just addition of matrix is just the addition of each element in the same position. For the matrix C, the element in the first row and first column is 1 plus 7, which is 8. In the same way, so you can formulate matrix C. How about multiplication of a number to the matrix? Here a number is what to scalar? Just number. If matrix A which is small a_ij and small c is a number, then c multiplied by A what? The ca_ij in a bracket, so it means that those number get into the bracket, get inside to the matrix for each element. So cA get the same order as matrix A. Order means that M by N matrix, that's the order of matrix. For example, if we want to multiply number minus 2 to the matrix 1, 2, 3, 4, we can write, minus 2 times 1, 2, 3, 4, like this with bracket equals what? It is minus two get into the matrix A each element, and it becomes minus 2, minus 4, minus 6, minus 8, like this. We write minus 1 times A as minus A. By using the addition of matrix, so B minus A is what? B plus, minus matrix A. For example, let's say the B equals 2, 3, 4, 4, and matrix A equals 1, 2, 3, 2. B minus A is just in the right-hand side, just deduction like each element together. Let me introduce of a product of matrix, so from now is different from multiplication of number to the matrix. How can you define the product of matrix? There's two matrix, A equals a_ij and matrix B equals b_ij, M by N and P by Q matrix respectively. A is M by N matrix, so order of A is what? M by N. B is matrix P by Q matrix, so order of B is P by Q. If the N equals P, which means the column or number of column in matrix A is equals to the number of row in the matrix B, then we can formulate the product of matrix AB. This is very important like the characteristics. If we want to product two matrix together, then we should check whether the first one, the left-hand, like a matrix, the number of column is equal to the rows of the second one against the B, should be matching to each other. If the number is different, then we cannot formulate the product of matrix. If product of matrix A times B is denoted by C, which is c_ij, then in this case C is M by Q matrix and the element of c_kp is calculated using the case law of A and p-th column or B as follow. If we want to calculate the element of C, for example, c_kp, we can use case law of A and B's column will be as follow. kp we can use a_k1 to a_kn times b_1p, b_2p to b_np like this. So a_k1 times b_1p is here, first calculate it, and plus a_k2 plus b_2p right there. Then up to the a_kN times b_Np. In a mathematical formulation, this is summation of N equals one to N, a_kn times b_kp like this. But this is a little bit complicated here. For the next session, I'll introduce with our example for the product of matrix. Thank you very much.
