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Math for CSLecture 21 Solution of Linear Systems of Equations Consistency Rank Geometric Interpretation Gaussian Elimination Lecture 2. Contents

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Math for CSLecture 22 Systems of linear equations In Linear equations the unknowns appear raised to the power one. If the variables in the equation are ordered as x 1, … x n and the missing variables x i are written as … + 0·x i + …, then the linear equation can be written in the matrix notation, according to the matrix multiplication rules. Solution of linear equations was one of the reasons, for matrix notation invention.

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Math for CSLecture 23 Linear Systems in Matrix Form (1)

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Math for CSLecture 24 Each side of the equation Can be multiplied by A -1 : Due to the definition of A -1 : Therefore the solution of (2) is: (2) Solution of Linear Systems

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Math for CSLecture 25 Consistency (Solvability) A -1 does not exist for every A The linear system of equations A · x=b has a solution, or said to be consistent IFF Rank{A}=Rank{A|b} A system is inconsistent when Rank{A}<Rank{A|b} Rank{A} is the maximum number of linearly independent columns or rows of A. Rank can be found by using ERO (Elementary Row Oparations) or ECO (Elementary column operations).

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Math for CSLecture 26 Elementary row and column operations The following operations applied to the augmented matrix [A|b], yield an equivalent linear system –Interchanges: The order of two rows/columns can be changed –Scaling: Multiplying a row/column by a nonzero constant –Sum: The row can be replaced by the sum of that row and a nonzero multiple of any other row. One can use ERO and ECO to find the Rank as follows: ERO minimum # of rows with at least one nonzero entry or ECO minimum # of columns with at least one nonzero entry

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Math for CSLecture 27 An inconsistent example: Geometric interpretation Rank{A}=1 Rank{A|b}=2 > Rank{A} ERO:Multiply the first row with -2 and add to the second row

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Math for CSLecture 28 Uniqueness of solutions The system has a unique solution IFF Rank{A}=Rank{A|b}=n n is the order of the system Such systems are called full-rank systems

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Math for CSLecture 29 Full-rank systems If Rank{A}=n Det{A} 0 A -1 exists Unique solution

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Math for CSLecture 210 Rank deficient matrices If Rank{A}=m<n Det{A} = 0 A is singular so not invertible infinite number of solutions (n-m free variables) under-determined system Consistent so solvable Rank{A}=Rank{A|b}=1

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Math for CSLecture 211 Ill-conditioned system of equations A small deviation in the entries of A matrix, causes a large deviation in the solution.

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Math for CSLecture 212 Ill-conditioned continued..... A linear system of equations is said to be “ill- conditioned” if the coefficient matrix tends to be singular

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Math for CSLecture 213 Gaussian Elimination –By using ERO, matrix A is transformed into an upper triangular matrix (all elements below diagonal 0) –Back substitution is used to solve the upper- triangular system ERO Back substitution

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Math for CSLecture 214 Pivotal Element Pivotal element The first coefficient of the first row (pivot) is used to zero out first coefficients of Other rows.

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Math for CSLecture 215 First step of elimination First row, multiplied by appropriate factor is subtracted from other rows.

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Math for CSLecture 216 Second step of elimination Pivotal element The second coefficient (pivot) of the second row is used to zero out second coefficients of other rows.

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Math for CSLecture 217 Second step of elimination Second row, multiplied by appropriate factor is subtracted from other rows (ERO).

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Math for CSLecture 218 Gaussion elimination algorithm For c=p+1 to n

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Math for CSLecture 219 Back substitution algorithm Finally, the following system is obtained:

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Math for CSLecture 220 Back substitution algorithm The answer is obtained as following:

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