{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# <span style=\"color:green\"><center>Diplomado en Ciencia de Datos</center></span>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# <span style=\"color:red\"><center>Espacios vectoriales en los reales</center></span>\n",
"## <span style=\"color:red\"><center>Vectores, matrices, norma de un vector, geometría</center></span>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<figure>\n",
"<center>\n",
"<img src=\"../Imagenes/Espacio-vectorial.png\" width=\"400\" height=\"400\" align=\"center\"/>\n",
"<figcaption>Espacio vectorial</figcaption>\n",
"</center>\n",
"</figure>\n",
"\n",
"Fuente: [Wikipedia](https://es.wikipedia.org/wiki/Espacio_vectorial)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## <span style=\"color:blue\">Profesores</span>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. Campo Elías Pardo Turriago, cepardot@unal.edu.co \n",
"2. Alvaro Mauricio Montenegro Díaz, ammontenegrod@unal.edu.co\n",
"3. Daniel Mauricio Montenegro Reyes, dextronomo@gmail.com \n",
"4. Camilo José Torres Jiménez, Msc, cjtorresj@unal.edu.co"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## <span style=\"color:blue\">Asesora Medios y Marketing digital</span>\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"4. Maria del Pilar Montenegro, pmontenegro88@gmail.com "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## <span style=\"color:blue\">Asistentes</span>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. Jessica López Mejia, jelopezme@unal.edu.co\n",
"2. Daniel Andrés Rojas, anrojasor@unal.edu.co"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Introducción"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"En esta lección aprenderemos los conceptos básicos de *espacios vectoriales en los reales con producto interno*. Un espacio vectorial está constinuido por el conjunto de vectores (número infinito) de la misma dimesión. Los múmeros reales se pueden ver como vectores de una dimensión y se pueden representar en una recta, que denominamos la *recta real*. El conjunto de todos los números reales conforman us espacio vectorial y se nota $Re$. Dos números *x* y *y* pertenecen a $Re$ y se nota, $x \\in \\Re$, $y \\in \\Re$. \n",
"\n",
"Las parejas ordenadas (si se cambia el orden se trata de una pareja distinta) de números reales, son vectores en $Re^2$, que son las infinitas parejas de dos números reales. Formalmente, la representación geométrica requiere de una *métrica* (*distancia*) que se define por medio de un *producto interno*. La geometría que conocemos desde la formación básica es una de ellas y podemos representar los vectores en $\\Re^2$ como puntos en un *plano cartesiano*, definido por dos rectas perpendiculares y horizontal y la otra vertical, las dos perpendiculares. La generalización matemática de estos conceptos a cualquier dimensión $n$ es el conjunto infinito de vectores definidos en $\\Re^n$.\n",
"\n",
"Laa *Geometría Euclidiana Multidimensional* se soporta entonces en *espacios vectoriales en $\\Re^n$ dotados de un producto interno*. Utilizaremos estos conceptos en las descripciones multivariadas de datos.\n",
"\n",
"Para facilitar a quienes no han abordado el Álgebra Lineal, empezaremos con los vectores en $\\Re^2$ y luego generalizaremos a $\\Re^n$.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Vectores en $\\Re^2$. Álgebra de vectores"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Un vector en $\\Re^2$ es una dupla de números reales. Sean **x** y **y** dos vectores en $\\Re^2$: $\\mathbf{x}=(x_1,x_2)$ y $\\mathbf{y}=(y_1,y_2)$ por ejemplo $\\mathbf{x} = (1,2)$ y $\\mathbf{y} = (1,3)$. Se definen dos operaciones: suma y la multiplicación por escalar.\n",
"- Suma: $\\mathbf{x} + \\mathbf{y} = (x_1+y_1,x_2+y_2)$. Ejemplo: $(1,2)+(1,3)=(1+1,2+3) = (2,5)$\n",
"\n",
"- Multiplicación por escalar: $c \\mathbf{x} = (cx_1,cx_2)$. Ejemplo: sea $c=3$, $3 \\mathbf{x} = 3(1,2) = (3,6)$\n",
"\n",
"Se dice que $\\Re^2$ dotado dotado de esas dos operaciones en un **espacio vectorial**, las duplas se llaman vectores y los elementos de las duplas se llaman componentes del vector. En la siguiente imagen (tomada del cuaderno sobre tensores) podemos reconocer varios vectores en $\\Re^2$:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"En una matriz $2 \\times 2$, como la del ejemplo, se pueden ver:\n",
"- dos vectores fila en $\\Re^2$: $(1,2)$ y $(3,4)$, \n",
"- dos vectores columna en $\\Re^2$: $(1,3)$ y $(2,4)$.\n",
"\n",
"En el tensor del ejemplo se pueden ver 4 vectores en $\\Re^2$: $(1,2)$, $(3,2)$, $(1,7)$, $(5,4)$. \n",
"\n",
"\n",
"En Estadística escribimos los vectores como columnas, es decir como matrices, en $\\Re^2$ ($2 \\times 1$).\n",
"Con esta opción estos 4 vectores son:\n",
"\n",
"$$\n",
"\\begin{bmatrix}\n",
" 1 \\\\\n",
" 2 \n",
"\\end{bmatrix}, \n",
"\\begin{bmatrix}\n",
" 3 \\\\\n",
" 2 \n",
"\\end{bmatrix}, \n",
"\\begin{bmatrix}\n",
" 1 \\\\\n",
" 7 \n",
"\\end{bmatrix}, \n",
"\\begin{bmatrix}\n",
" 5 \\\\\n",
" 4 \n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
" Entonces los dos vectores fila de la matriz, los vemos como dos vectores transpuestos. La transpuesta de una matriz ($ \\ \\ ^{'}$, $^t$ o $^T$) se obtiene intercambiando filas por columnas.\n",
"\n",
"Sea $\\mathbf{A} = \\begin{bmatrix}\n",
" 1 & 2 \\\\\n",
" 3 & 4\n",
"\\end{bmatrix}$ entonces la transpuesta de **A** es $\\mathbf{A}^T = \\begin{bmatrix}\n",
" 1 & 3 \\\\\n",
" 2 & 4\n",
"\\end{bmatrix}$\n",
"\n",
"El transpuesto de un vector (columna) es el mismo vector pero escrito como una fila.\n",
"\n",
"En esta sección hemos escrito unos cuantos vectores distintos, si los donotemos como $\\mathbf{x}_i, \\ i=1,2,\\cdots$\n",
"\n",
"$$\n",
"\\mathbf{x}_1=\\begin{bmatrix}\n",
" 1 \\\\\n",
" 2\n",
"\\end{bmatrix}, \\ \\mathbf{x}_2=\\begin{bmatrix}\n",
" 1 \\\\\n",
" 3\n",
"\\end{bmatrix}, \\ \\mathbf{x}_3=\\begin{bmatrix}\n",
" 3 \\\\\n",
" 4\n",
"\\end{bmatrix}, \\ \\mathbf{x}_4=\\begin{bmatrix}\n",
" 3 \\\\\n",
" 2\n",
"\\end{bmatrix},\\ \\mathbf{x}_5=\\begin{bmatrix}\n",
" 1 \\\\\n",
" 7\n",
"\\end{bmatrix},\\ \\mathbf{x}_6=\\begin{bmatrix}\n",
" 5 \\\\\n",
" 4\n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"Podemos sumar los 6 vectores:\n",
"$$\n",
"\\sum_{i=1}^6 \\mathbf{x}_i = \\mathbf{x}_1+\\mathbf{x}_2 + \\mathbf{x}_3 + \\mathbf{x}_4 + \\mathbf{x}_5 + \\mathbf{x}_6 =\n",
" \\begin{bmatrix} 1 \\\\\n",
" 2\n",
"\\end{bmatrix} + \\begin{bmatrix}\n",
" 1 \\\\\n",
" 3\n",
"\\end{bmatrix} + \\begin{bmatrix}\n",
" 3 \\\\\n",
" 4\n",
"\\end{bmatrix} + \\begin{bmatrix}\n",
" 3 \\\\\n",
" 2\n",
"\\end{bmatrix} + \\begin{bmatrix}\n",
" 1 \\\\\n",
" 7\n",
"\\end{bmatrix} + \\begin{bmatrix}\n",
" 5 \\\\\n",
" 4\n",
"\\end{bmatrix} = \n",
"\\begin{bmatrix} \n",
" 1+1+3+3+1+5 \\\\\n",
" 2+3+4+2+7+4\n",
"\\end{bmatrix} = \n",
"\\begin{bmatrix}\n",
" 14 \\\\\n",
" 22\n",
"\\end{bmatrix} \n",
"$$\n",
"\n",
"Si multiplicamos el vector suma por $\\dfrac{1}{6}=0.167$ obtenemos el vector promedio de los 6 vectores:\n",
"$$\n",
"\\dfrac{1}{6}\\ \\begin{bmatrix}\n",
" 14 \\\\\n",
" 22\n",
"\\end{bmatrix} = \\begin{bmatrix}\n",
" 2.333 \\\\\n",
" 3.667\n",
"\\end{bmatrix} \n",
"$$\n",
"Hemos utilizado las dos operaciones básicas de los vectores: suma y multiplicación por escalar, que forman parte de la definición de espacio vectorial."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"En R los podemos organizar en una matriz $2 \\times 6$ y verificar los resultados, así:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"La matrix X y sus vectores suma y promedio son:"
]
},
{
"data": {
"text/html": [
"<table class=\"dataframe\">\n",
"<caption>A matrix: 2 × 8 of type dbl</caption>\n",
"<thead>\n",
"\t<tr><th scope=col>x1</th><th scope=col>x2</th><th scope=col>x3</th><th scope=col>x4</th><th scope=col>x5</th><th scope=col>x6</th><th scope=col>Suma</th><th scope=col>Promedio</th></tr>\n",
"</thead>\n",
"<tbody>\n",
"\t<tr><td>1</td><td>1</td><td>3</td><td>3</td><td>1</td><td>5</td><td>14</td><td>2.333333</td></tr>\n",
"\t<tr><td>2</td><td>3</td><td>4</td><td>2</td><td>7</td><td>4</td><td>22</td><td>3.666667</td></tr>\n",
"</tbody>\n",
"</table>\n"
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"text/latex": [
"A matrix: 2 × 8 of type dbl\n",
"\\begin{tabular}{llllllll}\n",
" x1 & x2 & x3 & x4 & x5 & x6 & Suma & Promedio\\\\\n",
"\\hline\n",
"\t 1 & 1 & 3 & 3 & 1 & 5 & 14 & 2.333333\\\\\n",
"\t 2 & 3 & 4 & 2 & 7 & 4 & 22 & 3.666667\\\\\n",
"\\end{tabular}\n"
],
"text/markdown": [
"\n",
"A matrix: 2 × 8 of type dbl\n",
"\n",
"| x1 | x2 | x3 | x4 | x5 | x6 | Suma | Promedio |\n",
"|---|---|---|---|---|---|---|---|\n",
"| 1 | 1 | 3 | 3 | 1 | 5 | 14 | 2.333333 |\n",
"| 2 | 3 | 4 | 2 | 7 | 4 | 22 | 3.666667 |\n",
"\n"
],
"text/plain": [
" x1 x2 x3 x4 x5 x6 Suma Promedio\n",
"[1,] 1 1 3 3 1 5 14 2.333333\n",
"[2,] 2 3 4 2 7 4 22 3.666667"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"X <- matrix(c(1,2,1,3,3,4,3,2,1,7,5,4),2,6)\n",
"colnames(X)=paste(\"x\",1:6,sep=\"\")\n",
"cat(\"\\nLa matrix X y sus vectores suma y promedio son:\");cbind(X,Suma=rowSums(X),Promedio=1/6*rowSums(X))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Combinación lineal\n",
"\n",
"Si se tienen $m$ vectores, el resultado de multiplicar cada uno de ellos por una constante (en general distinta) y sumar, se denomina combinación lineal.\n",
"$$\n",
"\\sum_{i=1}^m c_i \\mathbf{x}_i = c_1x_1+c_2x_2+\\cdots+c_ix_i+\\cdots+c_mx_m\n",
"$$\n",
"\n",
"En el ejemplo de los 6 vectores, sean los $c_i$ $1, 2, 3, 4, 5, 6$\n",
"$$\n",
"\\sum_{i=1}^6 c_i \\mathbf{x}_i = 1\\times\\begin{bmatrix}\n",
" 1 \\\\\n",
" 2\n",
"\\end{bmatrix} \\ + 2\\times\\begin{bmatrix}\n",
" 1 \\\\\n",
" 3\n",
"\\end{bmatrix} \\ + 3\\times\\begin{bmatrix}\n",
" 3 \\\\\n",
" 4\n",
"\\end{bmatrix} \\ +4\\times\\begin{bmatrix}\n",
" 3 \\\\\n",
" 2\n",
"\\end{bmatrix} \\ + 5\\times\\begin{bmatrix}\n",
" 1 \\\\\n",
" 7\n",
"\\end{bmatrix} \\ + 6\\times\\begin{bmatrix}\n",
" 5 \\\\\n",
" 4\n",
"\\end{bmatrix} = \\begin{bmatrix}\n",
" 1+2+9 +12+5 +30\\\\\n",
" 2+6+12+8 +35+24\n",
"\\end{bmatrix} = \\begin{bmatrix}\n",
" 59 \\\\\n",
" 87\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. Haciendo la suma con un ciclo:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<style>\n",
".list-inline {list-style: none; margin:0; padding: 0}\n",
".list-inline>li {display: inline-block}\n",
".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
"</style>\n",
"<ol class=list-inline><li>59</li><li>87</li></ol>\n"
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"text/latex": [
"\\begin{enumerate*}\n",
"\\item 59\n",
"\\item 87\n",
"\\end{enumerate*}\n"
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"text/markdown": [
"1. 59\n",
"2. 87\n",
"\n",
"\n"
],
"text/plain": [
"[1] 59 87"
]
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"output_type": "display_data"
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],
"source": [
"c = 1:6\n",
"sum = c(0,0)\n",
"for (i in c )\n",
" sum = sum+c[i]*X[,i]\n",
"sum "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Nótese que la suma de los vectores es una combinación lineal con todos los $c_i$ iguales a 1. Y la media otra, con todos \n",
"los $c_i = \\frac{1}{6}."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Independencia lineal\n",
"\n",
"Cuando se tiene un conjunto de vectores, sin que ninguno de ellos sea combinación lineal de los otros, se dice que los vectores son linealmente independientes.\n",
"\n",
"En $\\Re^2$ solo se pueden tener parejas de vectores linealmente independientes. Si se tienen tres vectores, por ejemplo, siempre uno de ellos se puede expresar como combinación lineal de los otros dos, si son linealmente independientes por parejas.\n",
"\n",
"La dependencia lineal en $\\Re^2$ de una pareja de vectores se detecta si uno de ellos es igual al otro por una constante, es decir que son colineales."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Base y base canónica\n",
"\n",
"Cualquier pareja de vectores linealmente independientes es una base para $\\Re^2$, la base más sencilla se llama base canónica.\n",
"\n",
"La **base canónica** el espacio vectorial $\\Re^2$ es: $\\mathbf{e}_1 = \\begin{bmatrix}\n",
" 1 \\\\\n",
" 0\n",
"\\end{bmatrix},\\mathbf{e}_2 = \\begin{bmatrix}\n",
" 0 \\\\\n",
" 1\n",
"\\end{bmatrix} $"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"En este caso los coeficientes para obtener un vector a partir del los dos son los mismos componentes del vector, por ejemplo:\n",
"$$\n",
" \\begin{bmatrix}\n",
" 5 \\\\\n",
" 4\n",
"\\end{bmatrix} = 5 \\begin{bmatrix}\n",
" 1 \\\\\n",
" 0\n",
"\\end{bmatrix} + 4 \\begin{bmatrix}\n",
" 0 \\\\\n",
" 1\n",
"\\end{bmatrix} \n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Todos estos conceptos tienen sentido en matemáticas \n",
"como objetos abstractos. Sin embargo se pueden entender más fácilmente si le\n",
"asociamos una representación geométrica. Para formalizar la representación\n",
"geométrica "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Distancia o métrica en $\\Re^2$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"tags": []
},
"outputs": [
{
"data": {
"image/png": 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b9iCDG6YHKYOYNIUAAggg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\nTUlAc/d9AAAAAElFTkSuQmCC",
"text/plain": [
"Plot with title “Distancia euclidiana”"
]
},
"metadata": {
"image/png": {
"height": 420,
"width": 420
}
},
"output_type": "display_data"
}
],
"source": [
"# código para hacer la gráfica\n",
"# Rojas D.A y Pardo C.E\n",
"# no es necesario entenderlo\n",
"x1 <- 2\n",
"y1 <- 3\n",
"x2 <- 5\n",
"y2 <- 7\n",
"dist_euclidiana <- sqrt((x2 - x1)^2 + (y2 - y1)^2)\n",
"plot(c(x1, x2), c(y1, y2), type = \"n\", xlab = \"X\", ylab = \"Y\",\n",
" xlim=c(-0,5), ylim=c(0,8)#,main = \"Distancia Euclidiana en R2\"\n",
" ,asp=1,las=1,main=\"Distancia euclidiana\"\n",
" )\n",
"abline(v=-1:6,h=0:8,col=\"gray80\") # grilla\n",
"abline(v=0,h=0,col=\"darkgreen\") # ejes\n",
"points(x1, y1, col = \"red\", pch = 16)\n",
"points(x2, y2, col = \"blue\", pch = 16)\n",
"segments(x1, y1, x2, y2, col = \"black\", lty = 2)\n",
"text(x1, y1, labels = paste(\"(\", x1, \",\", y1, \")\", sep = \"\"), pos = 4)\n",
"text(x2, y2, labels = paste(\"(\", x2, \",\", y2, \")\", sep = \"\"), pos = 2)\n",
"text((x1 + x2) / 2, (y1 + y2) / 2, labels = sprintf(\"Distancia = %.2f\", dist_euclidiana), pos = 3)\n",
"segments(x1, y1, x1, y2, col = \"red\", lty = 3) \n",
"segments(x1, y2, x2, y2, col = \"blue\", lty = 3) \n",
"segments(x1, y1, x2, y2, col = \"green\", lty = 1) \n",
"text(x1 - 0.2, (y1 + y2) / 2, labels = \"Altura\", pos = 4, col = \"red\")\n",
"text((x1 + x2) / 2, y2 + 0.2, labels = \"Base\", pos = 3, col = \"blue\")\n",
"text((x1 + x2) / 2, (y1 + y2) / 2, labels = \"Hipotenusa\", pos = 1, col = \"green\")\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Nótese que en campo se pueden establecer ángulos rectos, por ejemplo con tres varas de 3, 4, 5 metros, respectivamiente."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Sean los vectores $\\mathbf{x}$ y $\\mathbf{y}$ de $\\Re^2$, se definen:\n",
"- **el producto escalar** de $\\mathbf{x}$ y $\\mathbf{y}$: $<\\mathbf{x},\\mathbf{y}> = \\sum\\limits_{i=1}^2 x_i y_i = x_1 y_1 + x_2 y_2$\n",
"\n",
"el producto escalar $<\\mathbf{x},\\mathbf{x}> $ es $\\sum\\limits_{i=1}^2 x_i^2 = x_1^2 + x_2^2$\n",
"\n",
"- **la norma de x**: $||x||=\\sqrt{<\\mathbf{x},\\mathbf{x}>} $\n",
"\n",
"- **la distancia entre x y y**: $d(\\mathbf{x},\\mathbf{y}) = ||\\mathbf{x}-\\mathbf{y}||$ \n",
"\n",
"- **el coseno entre x y y**: $cos(\\mathbf{x},\\mathbf{y}) = \\dfrac{<\\mathbf{x},\\mathbf{y}>}{||x||||y||}$\n",
"\n",
"- **la ortogonalidad entre x y y**: $\\mathbf{x} \\bot \\mathbf{y} \\ si \\ <\\mathbf{x},\\mathbf{y}> = 0$ \n",
"\n",
"Ahora $\\Re^2$ está dotado de una métrica (distancia) Euclidiana y tenemos una repesentación gráfica para los vectores.\n",
"\n",
"Para los vectores de la base canónina su norma es un y el producto escalar (también se le dice producto punto entre los dos vectores es cero.\n",
"$$\n",
"||\\mathbf{e}_1|| = 1 \\ \\ \\ <\\mathbf{e}_1,\\mathbf{e}_2>=0\n",
"$$\n",
"\n",
"Entonces la base canónica es una **base ortonormal**\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Distancia o métrica en $\\Re^2$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<figure>\n",
"<center>\n",
"<img src=\"../Imagenes/distancia-euclidiana.png\" width=\"400\" height=\"400\" align=\"center\"/>\n",
"<figcaption>Distancia euclidiana $\\Re^2$</figcaption>\n",
"</center>\n",
"</figure>\n",
"\n",
"Fuente: [Wikipedia](https://es.wikipedia.org/wiki/Distancia_euclidiana)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Sean los vectores $\\mathbf{x}$ y $\\mathbf{y}$ de $\\Re^2$, se definen:\n",
"- **el producto escalar** de $\\mathbf{x}$ y $\\mathbf{y}$: $<\\mathbf{x},\\mathbf{y}> = \\sum\\limits_{i=1}^2 x_i y_i = x_1 y_1 + x_2 y_2$\n",
"\n",
"el producto escalar $<\\mathbf{x},\\mathbf{x}> $ es $\\sum\\limits_{i=1}^2 x_i^2 = x_1^2 + x_2^2$\n",
"\n",
"- **la norma de x**: $||x||=\\sqrt{<\\mathbf{x},\\mathbf{x}>} $\n",
"\n",
"- **la distancia entre x y y**: $d(\\mathbf{x},\\mathbf{y}) = ||\\mathbf{x}-\\mathbf{y}||$ \n",
"\n",
"- **el coseno entre x y y**: $cos(\\mathbf{x},\\mathbf{y}) = \\dfrac{<\\mathbf{x},\\mathbf{y}>}{||x||||y||}$\n",
"\n",
"- **la ortogonalidad entre x y y**: $\\mathbf{x} \\bot \\mathbf{y} \\ si \\ <\\mathbf{x},\\mathbf{y}> = 0$ \n",
"\n",
"Ahora $\\Re^2$ está dotado de una métrica (distancia) Euclidiana y tenemos una repesentación gráfica para los vectores.\n",
"\n",
"Para los vectores de la base canónina su norma es un y el producto escalar (también se le dice producto punto entre los dos vectores es cero.\n",
"$$\n",
"||\\mathbf{e}_1|| = 1 \\ \\ \\ <\\mathbf{e}_1,\\mathbf{e}_2>=0\n",
"$$\n",
"\n",
"Entonces la base canónica es una **base ortonormal**\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Multiplicación matricial entre dos vectores en $\\Re^2$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<figure>\n",
"<center>\n",
"<img src=\"../Imagenes/Multi-matrices.png\" width=\"400\" height=\"400\" align=\"center\"/>\n",
"<figcaption>Multiplicación de matrices en $\\Re^2$</figcaption>\n",
"</center>\n",
"</figure>\n",
"\n",
"Fuente: [Wikipedia](https://es.wikipedia.org/wiki/Multiplicaci%C3%B3n_de_matrices)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Los vectores en $\\Re^2$, son también matrices de dimensión $2\\times 1$, así que veamos cómo se define la multiplicación de dos matrices.\n",
"\n",
"La matriz **A** se multiplica la matriz **B** y el resultado es la matriz **C**: \n",
"\n",
"Cada multiplicacion de matrices se hace por la fórmula fila (matriz **A**) * columna (matriz **B**)\n",
"\n",
"$ \\mathbf{A} = \\begin{bmatrix} 1 & 2\\\\ 3 & 4\\end{bmatrix}$, $\\mathbf{B} = \\begin{bmatrix} 5 & 10\\\\ 20 & 30\\end{bmatrix}$\n",
"\n",
"Entonces se tiene que \n",
"\n",
"$$\n",
"\\mathbf{AB} = \\begin{bmatrix} 1 & 2\\\\ 3 & 4\\end{bmatrix} \\begin{bmatrix} 5 & 10\\\\ 20 & 30\\end{bmatrix} = \\begin{bmatrix} 1\\times 5 + 2 \\times 20 & 1 \\times 10 + 2\\times 30\n",
"\\\\ 3\\times 5 + 4 \\times 20 & 3 \\times 10 + 4 \\times 30\\end{bmatrix} = \\mathbf{C}\n",
"$$\n",
"\n",
"Entonces los vectores en $\\Re^2$ se pueden multiplicar matricialmente de dos formas:\n",
"- $\\mathbf{x}^T \\mathbf{y} = [x_1 \\ \\ x_2 ] \\begin{bmatrix} y_1 \\\\ y_2 \\end{bmatrix} = x_1 y_1 + x_2 y_2$ que es el producto escalar entre **x** y **y**\n",
"\n",
"- $\\mathbf{x} \\mathbf{y}^T = \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} [y_1 \\ \\ y_2] = \\begin{bmatrix} x_1 y_1 & x_1 y_2 \\\\ x_2 y_1 & x_2 y_2 \\end{bmatrix}$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"El producto escalar entre dos vectores $\\mathbf{x}$ y $\\mathbf{x}$ en $\\Re^2$, se nota y define así:\n",
"\n",
"$$\n",
"<\\mathbf{x},\\mathbf{y}> = \\mathbf{x}^T \\mathbf{y} = x_1 y_1+x_2 y_2 = y_1 x_1 + y_2 x_2 = <\\mathbf{y},\\mathbf{x}> = \\mathbf{y}^T \\mathbf{x}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Los seis vectores son las columnas de la matriz **X**"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"X"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"t(X)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"El producto escalar $<\\mathbf{x}_1,\\mathbf{x}_2>$ es:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"as.numeric(t(X[,1])%*%X[,2])->x.y\n",
"cat(\"El producto escalar entre x,y es:\",x.y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Nótese que si multiplicamos la transpuesta de **X** por **X** obtenemos los productos internos entre todos los vectores:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"t(X)%*%X->prodsPunto\n",
"prodsPunto"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Las normas de los 6 vectores son las raíces cuadradas de la diagonal:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"normas=sqrt(diag(prodsPunto))\n",
"round(t(normas),2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Productos escalares:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"as.dist(prodsPunto)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Distancias: "
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"round(dist(t(X)),2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Representación geométrica en $\\Re^2$\n",
"\n",
"El espacio vectorial $\\Re^2$ dotado del producto interno clásico permite hacer la representación gráfica en el plano. El el eje horizontal se genera con el vector $mathbf{e}_1$ y el eje vertical son el vector $mathbf{e}_2$. Las dos filas de **X** las podemos llamar $V_1$ y $V_2$ respectivamente."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"tags": []
},
"outputs": [
{
"data": {
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LvkgKSKoWp7Jd9Mfp3U2+qqjV3WW+kO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\nBAgQWAGB/wFsL6PT2QEcIwAAAABJRU5ErkJggg==",
"text/plain": [
"plot without title"
]
},
"metadata": {
"image/png": {
"height": 420,
"width": 420
}
},
"output_type": "display_data"
}
],
"source": [
"rownames(X) = paste(\"V\",1:2,sep=\"\")\n",
"plot(t(X),las=1,xlim=c(0,5),ylim=c(0,7),asp=1)\n",
"text(t(X),colnames(X),pos=2) \n",
"abline(v=-1:6,h=0:7,col=\"gray80\") # grilla\n",
"abline(v=0,h=0,col=\"darkgreen\") # ejes\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Espacio vectorial $\\Re^n$ con producto interno clásico\n",
"\n",
"La generalización a cualquier espacio de dimensión $n$ es inmediata. Así se puede representan las tablas en espacios vectoriales. \n",
"\n",
"Sea $\\Re^n = \\{(x_1,x_2 \\cdots x^n) | x_i \\in \\Re,i=1,2 \\cdots n \\}$. En estadística se escriben en columnas $\\mathbf{y}$ y $\\mathbf{x}$:\n",
"$$\n",
"\\mathbf{x} =\\begin{bmatrix}\n",
" x_1 \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" x_i \\\\\n",
" \\cdot \\\\ \\cdot \\\\ \\cdot \\\\\n",
" x_n\n",
"\\end{bmatrix}\n",
"\\ \\ \\ \\ \\ \\ \\ \\ \\\n",
"\\mathbf{y} =\\begin{bmatrix}\n",
" y_1 \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" y_i \\\\\n",
" \\cdot \\\\ \\cdot \\\\ \\cdot \\\\\n",
" y_n\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Suma\n",
"\n",
"$$\n",
"\\mathbf{x} + \\mathbf{y} =\n",
"\\begin{bmatrix}\n",
" x_1 \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" x_i \\\\\n",
" \\cdot \\\\ \\cdot \\\\ \\cdot \\\\\n",
" x_n\n",
"\\end{bmatrix}\n",
"+ \n",
"\\begin{bmatrix}\n",
" y_1 \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" y_i \\\\\n",
" \\cdot \\\\ \\cdot \\\\ \\cdot \\\\\n",
" y_n\n",
"\\end{bmatrix} \n",
"=\n",
"\\begin{bmatrix}\n",
" x_1 + y_1 \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" x_i + y_i \\\\\n",
" \\cdot \\\\ \\cdot \\\\ \\cdot \\\\\n",
" x_n + y_n\n",
"\\end{bmatrix} \n",
"$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Multiplicación por escalar\n",
"\n",
"$$\n",
"c\\mathbf{x}\n",
"=\n",
"\\begin{bmatrix}\n",
" c x_1 \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" \\cdot \\\\\n",
" c x_i \\\\\n",
" \\cdot \\\\ \\cdot \\\\ \\cdot \\\\\n",
" c x_n\n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"El **espacio vectorial** $\\Re^n$ tiene dimensión $n$ y sus **vectores** $n$ componentes"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Combinación lineal en $\\Re^n$ \n",
"\n",
"Si se tienen $m$ vectores, el resultado de multiplicar cada uno de ellos por una constante (en general distinta) y sumar, se denomina combinación lineal.\n",
"$$\n",
"\\sum_{i=1}^m c_i \\mathbf{x}_i \n",
"$$\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Vectores linealmente independientes en $\\Re^n$\n",
"\n",
"Cuando se tiene un conjunto de vectores, sin que ninguno de ellos sea combinación lineal de los otros, se dice que los vectores son linealmente independientes.\n",
"\n",
"En $\\Re^n$ se pueden tener hasta $n$ de vectores linealmente independientes. Si se tienen $n+1$ vectores, por ejemplo, siempre uno de ellos se puede expresar como combinación lineal de los otros $n$ vectores que son linealmente independientes.\n",
"\n",
"Un conjunto de $n$ vectores linealmente independiente es una **base** para el espacio vectorial $\\Re^n$, se dice que la **dimensión del espacio vetorial** es $n$. \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Base canónica en $\\Re^n$\n",
"\n",
"La **base canónica** el espacio vectorial $\\Re^2$ es: \n",
"$$\n",
"\\mathbf{e}_1 = \\begin{bmatrix}\n",
" 1 \\\\\n",
" . \\\\ . \\\\ . \\\\\n",
" 0 \\\\\n",
" . \\\\ . \\\\ . \\\\\n",
" 0\n",
"\\end{bmatrix},\\mathbf{e}_2 = \\begin{bmatrix}\n",
" 0 \\\\\n",
" . \\\\ . \\\\ . \\\\\n",
" 1 \\\\\n",
" . \\\\ . \\\\ . \\\\\n",
" 0\n",
"\\end{bmatrix} \\cdots \\mathbf{e}_n = \\begin{bmatrix}\n",
" 0 \\\\\n",
" . \\\\ . \\\\ . \\\\ . \\\\ . \\\\ . \\\\ . \\\\\n",
" 1\n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"Obsérvese que la base canónica son las columnas de la **matriz identidad** $\\mathbf{I}_n$: matriz diagonal de unos $n \\times n$,\n",
"\n",
"$$\n",
"\\mathbf{I}_n = \\begin{bmatrix}\n",
" 1 & 0 & . & . & . & . & 0 \\\\\n",
" 0 & 1 & . & . & . & . & 0 \\\\\n",
" 0 & 0 & . & . & . & . & 0 \\\\\n",
" 0 & 0 & . & . & . & . & 0 \\\\\n",
" 0 & 0 & . & . & . & . & 0 \\\\\n",
" 0 & 0 & . & . & . & . & 0 \\\\\n",
" 0 & 0 & . & . & . & . & 1 \\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Subespacios vectoriales de $\\Re^n$\n",
"\n",
"Si cuando se tienen $q$ vectores linealmente independientes y si $q \\le n$ se dice que $\\Re^q$ es un **subespacio vectorial de** $\\Re^n$ si $q$ es igual a:\n",
"- 1 : líneas\n",
"- 2 : planos\n",
"- 3 : 3D\n",
"- q : $\\Re^q$\n",
"- n : $\\Re^n$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Producto escalar clásico, métrica y geometría multidimensional"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<figure>\n",
"<center>\n",
"<img src=\"../Imagenes/Scalar-product-dot-product.svg.png\" width=\"400\" height=\"400\" align=\"center\"/>\n",
"<figcaption>Representación del producto escalar en el espacio euclideo.</figcaption>\n",
"</center>\n",
"</figure>\n",
"\n",
"Fuente: [Wikipedia](https://es.wikipedia.org/wiki/Producto_escalar)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Sean los vectores $\\mathbf{x}$ y $\\mathbf{y}$ de $\\Re^n$, se definen:\n",
"- **el producto escalar** de $\\mathbf{x}$ y $\\mathbf{y}$: $<\\mathbf{x},\\mathbf{y}> = \\sum\\limits_{i=1}^n x_i y_i = \\mathbf{x}^T \\mathbf{y} $\n",
"\n",
"el producto escalar $<\\mathbf{x},\\mathbf{x}> $ es $\\sum\\limits_{i=1}^n x_i^n = \\mathbf{x}^T \\mathbf{x}$\n",
"\n",
"- **la norma de x**: $||x||=\\sqrt{<\\mathbf{x},\\mathbf{x}>} $\n",
"\n",
"- **la distancia entre x y y**: $d(\\mathbf{x},\\mathbf{y}) = ||\\mathbf{x}-\\mathbf{y}||$ \n",
"\n",
"- **el coseno entre x y y**: $cos(\\mathbf{x},\\mathbf{y}) = \\dfrac{<\\mathbf{x},\\mathbf{y}>}{||x||||y||}$\n",
"\n",
"- **la ortogonalidad entre x y y**: $\\mathbf{x} \\bot \\mathbf{y} \\ si \\ <\\mathbf{x},\\mathbf{y}> = 0$ \n",
"\n",
"Ahora $\\Re^n$ está dotado de una métrica (distancia) Euclidiana y tenemos una repesentación gráfica abstracta (cuando $n > 3$ para los vectores.\n",
"\n",
"Para los vectores de la base canónina su norma es un y el producto escalar o punto entre los dos vectores es cero.\n",
"$$\n",
"||\\mathbf{e}_i|| = 1 \\ \\ \\ <\\mathbf{e}_i,\\mathbf{e}_l>=0 \\ \\ \\forall i \\neq l\n",
"$$\n",
"\n",
"Entonces la base canónica es una **base ortonormal**\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Transformaciones lineales\n",
"\n",
"Una **transformación lineal** **T** es función lineal que va de un espacio vectorial en otro.\n",
"Se dice lineal porque cumple:\n",
"- $T(\\mathbf{x} + \\mathbf{y}) = T(\\mathbf{x}) +T( \\mathbf{y})$ \n",
"- $T(c \\mathbf{x}) = c T(\\mathbf{x})$.\n",
"\n",
"Hay una relación uno a uno entre transformaciones lineales y matrices, es decir:\n",
"- Toda transformación lineal tiene asociada una matriz\n",
"- Toda matriz define una transformación lineal."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Proyección ortogonal sobre un vector"
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"\n",
"<figure>\n",
"<center>\n",
"<img src=\"../Imagenes/Proyeccion1.png\" width=\"400\" height=\"400\" align=\"center\"/>\n",
"<figcaption>Proyección</figcaption>\n",
"</center>\n",
"</figure>\n",
"\n",
"Fuente: [Wikipedia](https://es.wikipedia.org/wiki/Producto_escalar)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Por ejemplo en $\\Re^2$ proyección del vector $\\mathbf{x}$ sobre el vector $\\mathbf{y}$\n",
"Sea $\\mathbf{P}$ la transformación y la matriz asociada:\n",
"$$\n",
"P: \\Re^2 \\longrightarrow \\Re\n",
"$$\n",
"\n",
"La matriz $\\mathbf{P} = \\mathbf{y}^T = [ y_1 \\ \\ y_2 ]$ y el resultado es:\n",
"$$\n",
"\\mathbf{y}^T \\mathbf{x}\n",
"$$\n",
"Nótese que su valor es igual a producto escalar entre **x** y **y**\n",
"$$\n",
"<\\mathbf{x},\\mathbf{y}> = \\mathbf{x}^T\\mathbf{y} = \\mathbf{y}^T \\mathbf{x}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Rotación en $\\Re^2$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"A: \\Re^2 \\longrightarrow \\Re^2\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\mathbf{A} =\\begin{bmatrix} \\cos \\theta & -\\sin \\theta \\\\\n",
"\\cos \\theta & \\sin \\theta \n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"- $\\theta$ representa el ángulo de rotación. Si En $\\theta = \\pi/4$ la rotación es de $45^o$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Diagonalización de una matriz: valores y vectores propios\n",
"\n",
"Una matriz $\\mathbf{V}$ define una transformación lineal de $\\Re^n$ en $\\Re^n$\n",
"$$\n",
"\\mathbf{V}: \\Re^n \\longrightarrow \\Re^n\n",
"$$\n",
"$\\mathbf{w} = \\mathbf{Vu}$\n",
"Cuando la transformación cumple \n",
"$$\n",
"\\mathbf{V} \\mathbf{u} = \\lambda \\mathbf{u}\n",
"$$\n",
"es decir $\\mathbf{w}=\\lambda \\mathbf{u}$ el vector $\\mathbf{u}$ multiplicado por la constante $\\lambda$,\n",
"se dice que $\\lambda$ es un valor propio de $\\mathbf{V}$ y $\\mathbf{u}$ es un vector propio asociado."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Ejercicio\n",
"construir un ejemplo en $\\Re^2$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Rango de una matriz\n",
"Si $\\mathbf{V}$ es una matriz definida positiva la matriz es de rango completo, es decir $n$.\n",
"Si es semidefinida positiva el rango es el número de valores propios mayores que 0."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"FIN Email: cepardot@unal.edu.co"
]
}
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