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L5_LinEqnDirect - MA2213 Lecture 5 Linear Equations(Direct...

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MA2213 Lecture 5 Linear Equations (Direct Solvers)
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Systems of Linear Equations p. 243-248 Occur in a wide variety of disciplines Mathematics Statistics Physics Chemistry Biology Economics Sociology Psychology Archaeology Geology Astronomy Anthropology Engineering Management Business Medicine Finance
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Matrix Form for a system of linear equations b x A = n n R A × n R b n R x coefficient matrix (solution) column vector (right) column vector
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Linear Equations in Mathematics Numerical Analysis Geometry Interpolation Least Squares Quadrature Algebra find intersection of lines or planes partial fractions 1 1 1 2 2 + + - = - x b x a x 0 = + b a 2 = - b a 1 = a 1 - = b Coefficient Matrix Vandermonde (for polyn. interp.) or Gramm Transpose of Vandermonde Lec 4 vufoil 13 (to compute weights) B B T = - 2 0 1 1 1 1 b a
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Matrix Arithmetic p. 248-264 Matrix Inverse Matrix Multiplication n m R A × 1 0 0 0 1 0 0 0 1 3 I Identity Matrix p n R B × p m R AB × 1 0 0 1 2 I - - - = - a c b d bc ad d c b a 1 1 Theorem 6.2.6 p. 255 A square matrix has an inverse iff (if and only if) its determinant is not equal to zero.
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Solution of (this means ) 0 det A exists and is unique. b A x A A b Ax 1 1 ) ( - - = = b A x A A 1 1 ) ( - - = b A x b A x I 1 1 - - = = multiplication is associative for nonsingular b x A = Proof A Remark In MATLAB use: x = A \ b;
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Column Rank of a Matrix Definition The column rank of a matrix } ,..., 1 , 0 { cr , m M R M n m × dimension of the subspace of 1 × m m R R ? 0 2 1 2 4 2 cr = spanned by the column vectors of = M cr Remark M maximal number of linearly independent column vectors M Question is the of
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Row Rank of a Matrix Definition The row rank of a matrix } ,..., 1 , 0 { rr , n M R M n m × is the dimension of the subspace of 1 × n R ? 0 2 1 2 4 2 rr = spanned by the row vectors of = M rr Remark M maximal number of linearly independent row vectors of M Question
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A Matrix Times a Vector = = n n nn n n n n b b b x x x a a a a a a a a a b x A 2 1 2 1 2 1 2 22 21 1 12 11 has solution iff b is a linear combination of columns of A = + + + n nn n n n n n b b b a a a x a a a x a a a x 2 1 2 1 2 22 12 2 1 21 11 1 The equation
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Existence of Solution in General The linear equation b x A = has a solution if and only if A b A cr ] [ cr = A IS SINGULAR!
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