vectores_ortogonales_y_matrices_ortogonales.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "vectores ortogonales y matrices ortogonales.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyO8M22S9Riw+X08jAvoa9K6",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/Mgobeaalcoba/Deepnote_Repository/blob/main/vectores_ortogonales_y_matrices_ortogonales.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UVMbMsVyFb8j"
      },
      "source": [
        "Ser ortogonal siempre es en relación con otro vector. Y para que lo sean deben formar,ambos vectores, un angulo de 90°\n",
        "¿Como lo podemos calcular en Python? "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "X3uD166fFnEM"
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "W1_AQATfF5Ab",
        "outputId": "d68ecea1-0395-4bfc-db23-a47b84bc21cb"
      },
      "source": [
        "x = np.array([0,0,2,2])\n",
        "print(x)\n",
        "y= np.array([0,0,2,-2])\n",
        "print(y)"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0 0 2 2]\n",
            "[ 0  0  2 -2]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "6nY9I-XVGFoa",
        "outputId": "221c2fbd-7196-4a48-96f1-f05fff5211f4"
      },
      "source": [
        "plt.quiver([x[0],y[0]], # plt.quiver es para graficar vectores\n",
        "           [x[1],y[1]],\n",
        "           [x[2],y[2]],\n",
        "           [x[3],y[3]],\n",
        "           angles = \"xy\",\n",
        "           scale_units = \"xy\",\n",
        "           scale = 1)\n",
        "plt.xlim(-2,4)\n",
        "plt.ylim(-3,3)\n",
        "\n",
        "plt.show()"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "E_0HDLctGtLj"
      },
      "source": [
        "v1 = np.array([2,2])\n",
        "v2= np.array([2,-2])\n"
      ],
      "execution_count": 4,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Ox8tPjd9G1GC",
        "outputId": "2a460490-53df-4d94-d7c9-f50278a0f142"
      },
      "source": [
        "print(v1.dot(v2.T)) # Veo si el producto me da 0.\n",
        "print(v1.dot(v2)) # Prueba para ver diferencia entre el producto por el traspuesto o no"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0\n",
            "0\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2SeBMYZeHKeM"
      },
      "source": [
        "Entonces estos dos vectores efectivamente tienen un angulo de 90 grados y son ortogonales entre ellos. ¿Que significa que sean ortonormales? Que su norma es 1 para cada vector"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "U_NeeSTBHUB9",
        "outputId": "5e0f53d2-2f89-490a-80e0-5df1ee0d621a"
      },
      "source": [
        "print(np.linalg.norm(v1))\n",
        "print(np.linalg.norm(v2))"
      ],
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "2.8284271247461903\n",
            "2.8284271247461903\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hhkIbGxYHfjD"
      },
      "source": [
        "Para que sean ortonormales debería dividir a esos vectores por su norma. Pero probemos con otros que ya lo sean"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HiiiqoV3HkmL"
      },
      "source": [
        "v1 = np.array([1,0])\n",
        "v2 = np.array([0,-1])"
      ],
      "execution_count": 11,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "et_c9_qTHqzU",
        "outputId": "1a64fc21-f14b-492f-b4aa-546298017ec9"
      },
      "source": [
        "print(v1.dot(v2.T))"
      ],
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "R-qHpcS9HwOj",
        "outputId": "8e83b9f0-edae-4765-dc0e-972c6d17b119"
      },
      "source": [
        "print(np.linalg.norm(v1))\n",
        "print(np.linalg.norm(v2))"
      ],
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1.0\n",
            "1.0\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XE-lYSWAHxnM"
      },
      "source": [
        "Son ortonormales en este último caso. Es imposible tener n vectores mutuamente ortonormales en un espacio de dimension. En r2 por ej solo podemos tener 2 vectores ortogonales. Ya que deben ser todos los vectores mutuamente ortogonales, es decir, forman entre todos ellos angulos de 90 grados. "
      ]
    }
  ]
}