LINEAR ALGEBRA

2 BASIC CONCEPTS A Matrix is a rectangular arrays of numbers (real or complex) enclosed in brackets for instance The numbers a 11,…a mn are called elements. Horizontal lines are called rows or row vectors. Vertical lines are called columns or column vectors.

LINEAR ALGEBRA3 MATRIX – BASIC CONCEPTS A matrix with ‘m’ rows and ‘n’ columns is called m x n matrix (read m by n matrix). In a double script notation of matrix element, a jk, the first subscript denotes row and second subscript denoted column. Row Matrix has only one row –[a 1, a 2,….a n ] Column matrix has only one column. Square matrix has same no of rows and columns. Diagonal of square matrix is called principal diagonal.

LINEAR ALGEBRA4 MATRIX MULTIPLICATION Let A =[a jk ] be an mxn matrix and B=[b jk ] and rxp matrix. Then the product AB (in this order) is defined only when r=n and is the mxp matrix C=[c jk ]whose elements are c jk =a j1 b 1k + a j2 b 2k + …..+ a jn b nk = We see that c jk is the inner product (dot product) of the j-th row vector of the first matrix A, and the k-th column vector of the secodn matrix B.

LINEAR ALGEBRA5

6 MATRIX MULTIPLICATION

LINEAR ALGEBRA7 MATRIX MULTIPLICATION

LINEAR ALGEBRA8 SOME EXAMPLES

LINEAR ALGEBRA9 SYSTEM OF LINEAR EQUATION A system of m linear equations (or set of m simulatenous equations) in n unknowns x1, x2,…xn is a set of equations of the form The a jk are given numbers, which are called coefficients of the system. The b i are also given numbers.

LINEAR ALGEBRA10 LINEAR EQUATION Ax = b – Single vector equation where the coefficient matrix A =[a jk ] is the m x n matrix.

LINEAR ALGEBRA11 AUGMENTED MATRIX The following is the augmented matrix obtained by adding the b column vector to the A matrix.

LINEAR ALGEBRA12 GAUSS ELIMINATION First step – Elimination of x1 from the second, third and mth equation. The first equation is called the pivotal equation in the first step. Using it we eliminate x1 from second, third and mth equation. This is done by substracting –a 21 /a 11 times the first equation from the second equation. –a 31 /a 11 times the first equation from the third equation etc This gives a new set of equations –a 11 x 1 +a 12 x 2 +…+a 1m x n = b 1 – c 22 x 2 +….+c 2n x n = b 2 – c m2 x 2 +…c mn x n = b m

LINEAR ALGEBRA13 GAUSS ELIMINATION The second step is to eliminate x2 from the third…mth equation in a similar fashion as we did in first step. In the third step we eliminate x3 and so on. We then have a system of the form –a 11 x 1 +a 12 x 2 +…+a 1m x n = b 1 – c 22 x 2 +….+c 2n x n = b 2 – k rr x r +…+k rn x n = br – 0 = b r+1 –. – 0 = b m –where r  m

LINEAR ALGEBRA14 GAUSS ELIMINATION We have three possible cases –No solution if r < m and one of the numbers b r+1,…bm is not zero. –Precisely one solution if r=n and b r+1 …bm, if present are zero. –Infinitely many solution if r < n

LINEAR ALGEBRA15 Example