lec19_linalgebra

lec19_linalgebra -         Introduction to...

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Unformatted text preview:         Introduction to (reminder of?) matrices Solving systems of equations Exercise: Solve 3D linear system Exercise: Write your own brute force routine for finding roots   HW on roots and systems of eqns posted today (due next Tues.) University of Mississippi Dept. of Physics and Astronomy Phys 503, Dr. Gladden     Matrices are mathematical objects which are handy for representing certain types of data structures. Represented as a 2D grid of numbers with dimensions m x n.   Above is a 3 x 3 matrix, which is said to be square (m=n) a11 A = a21 a31 a12 a22 a32 a13 a23 a33 University of Mississippi Dept. of Physics and Astronomy Phys 503, Dr. Gladden   Identity matrix: Inverse of a matrix: I= ￿ 1 0 ￿ 0 1   A −1   Determinant of a matrix: A=I ￿ ￿ ab det = ab − cd cd ab 3 cd ￿ ￿ = ￿ 3a 3b 3c 3d   Multiplication by a scalar: ￿ University of Mississippi Dept. of Physics and Astronomy Phys 503, Dr. Gladden     Consider a system of N algebraic equations with N unknown quantities. How do we solve for these unknowns? 5x + 1y = 10   3x − 2y = 4   Solve for x in terms of y using eqn 1 and substitute into eqn 2. Then solve for y and use that to solve for x. Can represent this system as a matrix problem. University of Mississippi Dept. of Physics and Astronomy Phys 503, Dr. Gladden   In equation from, this would be AX = b ⇒     Now solve by multiplying inverse Need to be careful taking the inverse of a matrix – can lead to divide by 0, or singularities. ￿ 3 5 5x + 1y = 10 ￿￿ ￿ ￿ ￿ −2 x 4 = 1 y 10 A−1 AX = A−1 b IX = A −1 3x − 2y = 4 b University of Mississippi Dept. of Physics and Astronomy Phys 503, Dr. Gladden   Consider the following linear system −10 x + 10 y = 2 −10 x − y + 2 z = 4 2 x − 10 y − 8 z = −3   Express the problem in a matrix format in Python and solve X x0 , yo , zo University of Mississippi Dept. of Physics and Astronomy Phys 503, Dr. Gladden ...
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