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LINEAR ALGEBRA

LINEAR ALGEBRA. 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.

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LINEAR ALGEBRA

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  1. LINEAR ALGEBRA

  2. BASIC CONCEPTS • A Matrix is a rectangular arrays of numbers (real or complex) enclosed in brackets for instance • The numbers a11,…amn are called elements. • Horizontal lines are called rows or row vectors. • Vertical lines are called columns or column vectors. LINEAR ALGEBRA

  3. 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, ajk, the first subscript denotes row and second subscript denoted column. • Row Matrix has only one row • [a1, a2,….an] • Column matrix has only one column. • Square matrix has same no of rows and columns. • Diagonal of square matrix is called principal diagonal. LINEAR ALGEBRA

  4. MATRIX MULTIPLICATION Let A =[ajk] be an mxn matrix and B=[bjk] and rxp matrix. Then the product AB (in this order) is defined only when r=n and is the mxp matrix C=[cjk]whose elements are cjk=aj1b1k + aj2b2k + …..+ ajnbnk= We see that cjk 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 ALGEBRA

  5. LINEAR ALGEBRA LINEAR ALGEBRA

  6. MATRIX MULTIPLICATION LINEAR ALGEBRA

  7. MATRIX MULTIPLICATION LINEAR ALGEBRA

  8. SOME EXAMPLES LINEAR ALGEBRA

  9. 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 ajk are given numbers, which are called coefficients of the system. The bi are also given numbers. LINEAR ALGEBRA

  10. LINEAR EQUATION Ax = b – Single vector equation where the coefficient matrix A =[ajk] is the m x n matrix. LINEAR ALGEBRA

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

  12. 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 • a21/a11 times the first equation from the second equation. • a31/a11 times the first equation from the third equation etc • This gives a new set of equations • a11x1+a12x2+…+a1mxn = b1 • c22x2+….+c2nxn = b2 • cm2x2+…cmnxn = bm LINEAR ALGEBRA

  13. 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 • a11x1+a12x2+…+a1mxn = b1 • c22x2+….+c2nxn = b2 • krrxr+…+krnxn = br • 0 = b r+1 • . • 0 = bm • where r m LINEAR ALGEBRA

  14. 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 ALGEBRA

  15. Example LINEAR ALGEBRA

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