Intro to Linear Systems in Python.ipynb

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        "1. Solve the system of equations using np.linalg.inv(A).dot(b). You will need to define the matrix A and the vector b using the numpy array command. Then discuss the validity of your answer by solving the system exactly by hand and considering the double-precision arithmetic used by Python.\n",
        "\n",
        "x + 2y +  3z  = 3\n",
        "\n",
        "2x + 3y + 5z = 4\n",
        "\n",
        "3x + 4y + 5z = 5\n",
        "\n"
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        "import numpy as np\n",
        "\n",
        "\n"
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        "  2. Consider the following Python code and answer the following questions: \n",
        "  \n",
        "\n",
        "a.   Notice that the output of function $A$ is a matrix. What is the exact form of this matrix in terms of epsilon? What is the value of $A(1)$? $A(-1)$? $A(0)$?\n",
        "b.  What is the exact solution (in terms of $\\epsilon$) to:   $A(\\epsilon) x = b$ where $A(\\epsilon)$ and $b$ are as defined in the Python code below?\n",
        "c.  What is the determinant of $A(\\epsilon)$?\n",
        "d.  Explain exactly what is happening in the for loop in the code below?\n",
        "e.  Comment on the accuracy of the entire process and solution x as the loop progresses? If something's not correct/exact, what should the exact computation/solution be?\n",
        "\n",
        "\n"
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        "import numpy as np\n",
        "\n",
        "def A(epsilon):\n",
        " return np.array([[1,2],[1,2+epsilon]])\n",
        "     \n",
        "b = np.array([3,-2])  \n",
        "  \n",
        "for i in (0.1, 0.01, 0.001, 1e-6, 1e-12):\n",
        "    x = np.linalg.inv(A(i)).dot(b)\n",
        "    print(\"When epsilon is \", i, \" the solution to A(epsilon) x = b is x=\", x)"
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