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**Unformatted text preview: **MA2213 Lecture 5 Linear Equations (Direct Solvers) Systems of Linear Equations p. 243-248 ccur in a wide variety of disciplines Chemistry Biology Archaeology Astronomy Anthropology Business Mathematics Economics Geology Engineering Management Finance Statistics Physics Sociology Psychology Medicine 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 Linear Equations in Mathematics Numerical Analysis Coefficient Matrix Interpolation Vandermonde (for polyn. interp.) or Gramm B B T Least Squares Quadrature Transpose of Vandermonde Lec 4 vufoil 13 (to compute weights) Geometry find intersection of lines or planes 1 1 1 2 2 + + − = − x b x a x partial fractions Algebra 1 = a 1 − = b ⇒ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 2 1 1 1 1 b a = + b a 2 = − b a ⇒ Matrix Arithmetic p. 248-264 atrix Inverse atrix Multiplication n m R A × ∈ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ≡ 1 1 1 3 I Identity Matrix p n R B × ∈ p m R AB × ∈ ⇒ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ≡ 1 1 2 I ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − a c b d bc ad d c b a 1 1 heorem 6.2.6 p. 255 A square atrix has an inverse iff (if and only if) its determinant is not equal to zero. Solution of (this means ) det ≠ A xists 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 = roof A emark In MATLAB use: x = A \ b; Column Rank of a Matrix efinition The column rank of a matrix } ,..., 1 , { cr , m M R M n m ∈ ∈ × imension of the subspace of 1 × ≡ m m R R ? 2 1 2 4 2 cr = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ panned by the column vectors of = M cr emark M maximal number of linearly independent column vectors M uestion is the f Row Rank of a Matrix efinition The row rank of a matrix } ,..., 1 , { rr , n M R M n m ∈ ∈ × is the dimension of the subspace of 1 × n R ? 2 1 2 4 2 rr = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ panned by the row vectors of = M rr emark M maximal number of linearly independent row vectors of M uestion 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 M M L M O M M L L 2 1 2 1 2 1 2 22 21 1 12 11 as 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 M M L M M 2 1 2 1 2 22 12 2 1 21 11 1 he equation Existence of Solution in General The linear equation b x A = has a solution if and only if...

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