[MUSIC] Usually data is organized as arrays. The one-dimensional arrays, that is, lists or columns are called vectors and some rectangular ones are called matrices. Now, let us consider operations, which could be applied to them. An n-vector is just a tuple of n numbers, and they can be organized as a list or sum, but any tuple of n numbers is called n-vector. Maybe you know, that vectors are arrars or the vectors are for example in some programming languages a vector can be an array of numbers of any kind, but now in linear algebra n-vectors are arrays of n objects. Later we'll consider also abstract vectors, but let us concentrate now on these ones. For example, one-vector is just one number, so that the set of all one-vectors is the set of all real numbers, the n space is the set of all n-vectors, so that set of all n-tuples of numbers. So it is denoted by R^n as you know from the calculus course. The 2-vectors are pairs of numbers so that each 2-vector encode one point in the coordinate planes. We can identify our 2 with the coordinate plane. The 3-vectors are just usually vectors in their three dimensional space so you can imagine as some arrows as physical objects. Now, vectors form matrices. A matrix is a rectangular array of numbers, namely, an m by n matrix is an array with m rows and n columns so that it's an array (rectangular) with m by n cells, each cell contains a single number. The number which lies in i-th row and j-th column is denoted by aij, so that the first coordinate i, the second column j, and all coordinates are to from 1 to the corresponding sizes, to the number of rows and to the number of columns. For notation, look at the top: a sub ij with parentheses and the limits for the i and for the j are on the top, so the first index i, denotes the number of the row number and the second index j, denotes the column number not vice versa. For example, look at this formula: (i + j) and then very important to note that i is the first index, so that i is a row index and j is the second one, so that j is the column index. You see that the matrix has 2 rows because i has the limits 1 to 2 and j has limits 1 to 3. So that the matrix has three columns, and the elements which lie on the i-th row, and j-th column, is calculated by the formula inside the parentheses i + j, so that we get the matrix of this form, 2 by 3 matrix consisting of these numbers. Another example. Any number is a 1 by 1 matrix. The third example. Suppose that matrix B is given by the following formula. You see that again this formula gives us a way to calculate each element of the matrix. If you like programming, you can translate this formula to that function that calculates the elements on the is i-th and j-th cell. So that this matrix consists of single row, four columns. Each element of this matrix is calculated by the formula j square. Now, let us consider some special matrices. The row vector is a matrix of 1 by n, the column vector is a matrix n by 1, so each vector, n-vector can be represented by two matrices that are row vector and column vector. We will identify the n-vectors with the column vectors. So that Rn for us is the space of the column vectors. It maybe looks like something that doesn't important but later we will see that there are some points where the row vectors and the column vectors are not equivalent to each other. Analogously, the set of all m by n matrices is denoted by this R up m by n. So, similar to Rn the set of n vectors, while matrices are two-dimensional -- there are two dimensions. Another, one special matrix is the null or zero matrix, which consists of zeros (each cell is equal to 0). So for each size, n by n, we have one null matrix, which is sometimes denoted by both 0. So very special matrix is the null matrix. Now let us consider some basic operations, which can be applied to matrices. Some of them are arithmetic operations, and some of them are special for matrices and which can't be applied to number or at least if you apply them to numbers your doesn't have any new. Now, let A be an m by n matrix as before, remember that this notation means that the first elements is a11, the second one is a12, etc., up to m rows and n columns. So, the size of A, the size is a kind of operation! To each matrix, it associates a pair of its dimensions so that the first number is the number of rows, m, and the second number is the number of columns, n, so the size is a pair m and n. Sometimes we can put m times n. Now, another operation is cutting one row from the matrix, or extracting one row. The i-th row is denoted by Ai if the matrix itself is denoted by A, so the i-th row is the row vector. And the j-th column of A is the column vector, consisting of all elements of this column so that their row has the length n and the column has head m. Now, one important operation is transposition. The transpose of the matrix. If you have for example some matrix, you can just transpose it by this way. So you get from m by n matrix a matrix of the size n by m, the rows and columns are interchanged in the transpose matrix. So, it is given by the following formula, the first index in the transpose matrix is j, the second index is i. So that if the original matrix have had m rows and n columns, then the transpose will have m columns and n rows. For example, the row vector after transposition, gives the column vector and the column vector gives the row vector. So, the two kinds of vectors are transpose to each other. One another example. Recall that we have considered the matrix given by this formula. This is just a 2 by 3 matrix, 2, 3, 4, 3, 4, 5, now we have that the transpose of it is given by very similar formula but j and i are interchanged. Then we get the following matrix. From the long matrix, we get the high matrix, from the 2 by 3, we get the matrix of (size) 3 by 2. We rearranged same elements. So the first column was 2 by 3, so that the first row of the transpose matrix will be two by three, etc. Each column becomes now a row and vice versa. [MUSIC]