{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/financieras/pyCourse/blob/main/jupyter/calisto3/0100_numpy8_algebra_lineal.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "cdfe7257",
      "metadata": {
        "id": "cdfe7257"
      },
      "source": [
        "# NumPy [8] Álgebra lineal"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "fc0934ae",
      "metadata": {
        "id": "fc0934ae",
        "outputId": "1f983451-ec4d-43ed-a1f9-de57da5ca870"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[ 1  2  3]\n",
            " [-2  0  1]]\n"
          ]
        }
      ],
      "source": [
        "import numpy as np\n",
        "\n",
        "m = np.array([[1,2,3], [-2,0,1]])\n",
        "print(m)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "6736da85",
      "metadata": {
        "id": "6736da85",
        "outputId": "9a95994e-9607-48da-c384-23c39257d223"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[1]\n",
            " [2]\n",
            " [3]]\n"
          ]
        }
      ],
      "source": [
        "b = np.array([[1], [2], [3]])\n",
        "print(b)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "9d74d825",
      "metadata": {
        "id": "9d74d825"
      },
      "source": [
        "## Transposición matricial"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "63fa5b73",
      "metadata": {
        "id": "63fa5b73",
        "outputId": "efe0d4d7-54b9-4f6d-eec6-2af684b123a6"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[1 2 3]]\n"
          ]
        }
      ],
      "source": [
        "a = np.transpose(b)           # cambiamos filas por columnas\n",
        "print(a)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "bb407acc",
      "metadata": {
        "id": "bb407acc",
        "outputId": "ee5ed72a-45a9-4f81-a39b-c2974d155b98"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[ 1, -2],\n",
              "       [ 2,  0],\n",
              "       [ 3,  1]])"
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.transpose(m)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "5f3ea381",
      "metadata": {
        "id": "5f3ea381",
        "outputId": "c3c3ea33-4964-45a9-915e-a1e68b033a86"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[1]\n",
            " [2]\n",
            " [3]]\n"
          ]
        }
      ],
      "source": [
        "print(np.transpose(a))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "0f1aa5ef",
      "metadata": {
        "id": "0f1aa5ef"
      },
      "source": [
        "#### Comprobación\n",
        "Es necesario usar dobles corchetes."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "2f039e9a",
      "metadata": {
        "id": "2f039e9a",
        "outputId": "b8618b1e-e39f-404a-f312-fef0bc340af4"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[1]\n",
            " [2]\n",
            " [3]]\n"
          ]
        }
      ],
      "source": [
        "print(np.transpose([[1, 2, 3]]))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "156384c9",
      "metadata": {
        "id": "156384c9"
      },
      "source": [
        "## Resolución de sistemas lineales\n",
        "$$Ax=b$$\n",
        "Nuestro objetivo es calcular $x$.\n",
        "\n",
        "$$A=\n",
        "\\begin{pmatrix}\n",
        "1 & -3 &  2\\\\\n",
        "5 &  6 & -1\\\\\n",
        "4 & -1 &  3\n",
        "\\end{pmatrix}$$.\n",
        "\n",
        "$$b=\n",
        "\\begin{pmatrix}\n",
        "-3 \\\\\n",
        "13 \\\\\n",
        "8\n",
        "\\end{pmatrix}$$\n",
        "\n",
        "Solución:\n",
        "$$x=A^{-1}b=\n",
        "\\begin{pmatrix}\n",
        "-2 \\\\\n",
        "5 \\\\\n",
        "7\n",
        "\\end{pmatrix}$$\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "ff090b85",
      "metadata": {
        "id": "ff090b85",
        "outputId": "d6dd49e2-dbc3-43ac-aa6c-9c2439e774c6"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[ 1, -3,  2],\n",
              "       [ 5,  6, -1],\n",
              "       [ 4, -1,  3]])"
            ]
          },
          "execution_count": 15,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "A = np.array([[1,-3,2], [5,6,-1], [4,-1,3]])\n",
        "print(A)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "75c77ada",
      "metadata": {
        "id": "75c77ada",
        "outputId": "9ecdbc67-4389-4dbb-e8da-0555205e0ad0"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[-3]\n",
            " [13]\n",
            " [ 8]]\n"
          ]
        }
      ],
      "source": [
        "b = np.array([[-3], [13], [8]])\n",
        "print(b)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "0ab4fa44",
      "metadata": {
        "id": "0ab4fa44"
      },
      "outputs": [],
      "source": [
        "x = np.linalg.solve(A, b)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "283a3535",
      "metadata": {
        "id": "283a3535",
        "outputId": "ccbf5511-5a7f-45e5-f9df-962d66eee9fa"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[[-2.]\n",
            " [ 5.]\n",
            " [ 7.]]\n"
          ]
        }
      ],
      "source": [
        "print(x)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "b46c78de",
      "metadata": {
        "id": "b46c78de"
      },
      "source": [
        "#### Comprobación\n",
        "La función allclose nos permite comprobar que los valores, aunque no sean exactamente iguales, si son muy cercanos."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "1028a3a8",
      "metadata": {
        "id": "1028a3a8",
        "outputId": "6c2b6293-19b8-4449-a01b-cef11f1d4162"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "True"
            ]
          },
          "execution_count": 20,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.allclose(np.dot(A,x),b)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "07264f90",
      "metadata": {
        "id": "07264f90",
        "outputId": "f4b2a47b-e04b-4e14-91d8-8def9352bb61"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[-3.],\n",
              "       [13.],\n",
              "       [ 8.]])"
            ]
          },
          "execution_count": 22,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.dot(A, x)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "81524cfc",
      "metadata": {
        "id": "81524cfc",
        "outputId": "6ace7639-049e-4df4-dee7-9890b12a3cf0"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[-3],\n",
              "       [13],\n",
              "       [ 8]])"
            ]
          },
          "execution_count": 23,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "b"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "38ba4c93",
      "metadata": {
        "id": "38ba4c93",
        "outputId": "8b93efa5-fd99-4308-d546-f6c4e62aaa79"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[-3.55271368e-15],\n",
              "       [ 0.00000000e+00],\n",
              "       [ 3.55271368e-15]])"
            ]
          },
          "execution_count": 25,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.dot(A,x) - b"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "f7b0fc8c",
      "metadata": {
        "id": "f7b0fc8c"
      },
      "source": [
        "### Otra forma de cambiar las dimensiones de un vector\n",
        "En la práctica equivale a transponer el vector."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "2dd1de85",
      "metadata": {
        "scrolled": true,
        "id": "2dd1de85",
        "outputId": "2fe21248-b782-4cbf-c92a-90e4f1a875c4"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "[1 2 3]\n"
          ]
        }
      ],
      "source": [
        "c = np.array([1,2,3])\n",
        "print(c)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "485e7306",
      "metadata": {
        "id": "485e7306",
        "outputId": "546c2dd6-f566-4436-d23f-2429815bca22"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[1],\n",
              "       [2],\n",
              "       [3]])"
            ]
          },
          "execution_count": 38,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "c.shape = (3, 1)\n",
        "c"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "37006158",
      "metadata": {
        "id": "37006158",
        "outputId": "73e90158-0730-41ca-daa7-561d57855e8d"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[1, 2, 3]])"
            ]
          },
          "execution_count": 39,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "c.shape = (1, 3)\n",
        "c"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3 (ipykernel)",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.8.11"
    },
    "colab": {
      "name": "0100_numpy8_algebra_lineal.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}