{"id":32543,"date":"2022-03-08T13:00:42","date_gmt":"2022-03-08T13:00:42","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32543"},"modified":"2025-03-10T01:31:47","modified_gmt":"2025-03-10T01:31:47","slug":"the-euclidean-space-rn","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-euclidean-space-rn\/","title":{"rendered":"The Euclidean Space Rn"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>The Euclidean Space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^n}<\/span><\/span><\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\"><em>In this class, we explore the <strong>Euclidean space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/strong>, its algebraic structure, and metric properties. You will learn about vector operations, the <strong>dot product<\/strong>, the <strong>norm<\/strong>, and the <strong>Euclidean distance<\/strong>, essential concepts in geometry and analysis. With clear explanations and intuitive examples, this material will help you understand how space is mathematically modeled in multiple dimensions.<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<strong>Learning Objectives:<\/strong><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Define<\/strong> the Euclidean space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> and its fundamental properties.<\/li>\n<li><strong>Explain<\/strong> the vector structure of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> through its basic operations.<\/li>\n<li><strong>Apply<\/strong> the dot product to compute angles and projections between vectors.<\/li>\n<li><strong>Demonstrate<\/strong> algebraic and metric properties of the dot product in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>.<\/li>\n<li><strong>Use<\/strong> the Euclidean norm to determine the magnitude of a vector.<\/li>\n<li><strong>Calculate<\/strong> the Euclidean distance between two points in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> and analyze its geometric meaning.<\/li>\n<li><strong>Verify<\/strong> the validity of fundamental inequalities such as Cauchy-Schwarz and the triangle inequality.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">The Space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/a><br \/>\n<a href=\"#2\">The Dot Product<\/a><br \/>\n<a href=\"#3\">The Norm and the Euclidean Distance<\/a><br \/>\n<a href=\"#4\">Conclusion<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/mV-G69l9LtI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Vector Space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=123s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Surely before reaching this point, you were familiar with the properties of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> or the plane <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2,<\/span><\/span> or space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span><\/span><\/strong><\/a> All these ideas are useful for understanding the space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> Above all, the set <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n = \\{\\vec{x} = (x_1, \\cdots, x_n) | x_1, \\cdots, x_n \\in \\mathbb{R}\\},<\/span><\/span> equipped with the usual operations of vector addition and scalar multiplication, is a vector space. Let&#8217;s delve into this by reviewing the basic operations of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span><\/p>\n<h3>Basic operations in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=232s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n), \\vec{y}=(y_1, \\cdots, y_n)<\/span><\/span> are vectors in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> is any real scalar,<\/span><\/strong> <\/a>then the operations of <strong>vector addition<\/strong> and <strong>scalar multiplication<\/strong> are as described below:<\/p>\n<p><strong>Vector addition:<\/strong> Vector addition is described by the function:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcrl} +:&amp; \\mathbb{R}^n \\times \\mathbb{R}^n &amp; \\longrightarrow &amp; \\mathbb{R}^n \\\\ &amp; (\\vec{x},\\vec{y}) &amp; \\longmapsto &amp; \\vec{x}+\\vec{y} = (x_1+y_1, \\cdots, x_n + y_n) \\end{array} <\/span><\/span><\/p>\n<p><strong>Scalar multiplication:<\/strong> Scalar multiplication is described by the function:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcrl} ():&amp; \\mathbb{R} \\times \\mathbb{R}^n &amp; \\longrightarrow &amp; \\mathbb{R}^n \\\\ &amp; (\\alpha,\\vec{x}) &amp; \\longmapsto &amp; (\\alpha\\vec{x}) = (\\alpha x_1, \\cdots, \\alpha x_n) \\end{array} <\/span>\n<h3>Vector space properties of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=428s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">The space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> equipped with the operations described above<\/span><\/strong><\/a> is a <strong>vector space<\/strong> because its operations of addition and scalar multiplication satisfy the following properties:<\/p>\n<p>First, we have the <strong>commutative<\/strong> and <strong>associative<\/strong> properties.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{x} + \\vec{y} = \\vec{y} + \\vec{x}  \\\\ \\vec{x} + (\\vec{y}  + \\vec{z}) = (\\vec{x} + \\vec{y})  + \\vec{z}  \\\\ (\\alpha \\beta) \\vec{x}  = \\alpha (\\beta  \\vec{x}) = \\beta (\\alpha  \\vec{x}) = (\\beta\\alpha) \\vec{x}\n\n<\/span>\n<p><strong>The sum of scalars distributes over scalar multiplication, and vector addition distributes over scalar multiplication;<\/strong> that is, the following equalities hold:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha + \\beta) \\vec{x} = \\alpha\\vec{x} + \\beta\\vec{x} \\\\ \\alpha(\\vec{x} + \\vec{y}) = \\alpha\\vec{x} + \\alpha\\vec{y} <\/span>\n<p>There exists an <strong>additive identity<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{0}=(0,\\cdots, 0)<\/span> that satisfies the property:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{x} + \\vec{0} = \\vec{x} <\/span>\n<p>There exists a <strong>multiplicative identity<\/strong> for scalar multiplication:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> 1 \\vec{x} = \\vec{x} <\/span>\n<p>And every vector <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span> has an <strong>additive inverse<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\vec{x},<\/span><\/span> which satisfies the property:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{x} + -\\vec{x} = \\vec{0} <\/span><\/span><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HL85aSpHdsI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The Dot Product<\/h2>\n<p>When examining the construction of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> as a vector space, we notice that it lacks a multiplication operation between vectors; initially, we cannot \u00abmultiply\u00bb vectors as we do with real numbers. However, it is possible to define such an operation, and one way to do so is through what is known as the <strong>dot product.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=349s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">The dot product should not be confused with scalar multiplication,<\/span><\/strong><\/a> the former is a product between two vectors that yields a scalar, while the latter is the multiplication of a scalar by a vector, resulting in another vector. Consider two vectors in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n:<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n)<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1, \\cdots, y_n).<\/span><\/span> The dot product of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y},<\/span><\/span> denoted as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y},<\/span><\/span> is defined as the real number given by the formula:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} =\\displaystyle \\sum_{i=1}^n x_i y_i = x_1y_1 + \\cdots x_ny_n<\/span>\n<p>There are many ways to represent the dot product of vectors in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n,<\/span><\/span> one being the formula above. Another is obtained by considering a basis of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> and using the <strong>Einstein summation convention:<\/strong> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{e}_i\\}_{i=\\overline{1,n}}<\/span><\/span> is a basis of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> (usually the canonical basis), then the vectors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span><\/span> can be written as:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=\\displaystyle\\sum_{i=1}^n x_i\\hat{e}_i = x_1\\hat{e}_1 + \\cdots x_n\\hat{e}_n<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=\\displaystyle\\sum_{i=1}^n y_i\\hat{e}_i = y_1\\hat{e}_1 + \\cdots y_n\\hat{e}_n<\/span>\n<p>This explicitly indicates that the coefficients <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y_i<\/span><\/span> of the vectors are relative to the basis of the space.<\/p>\n<h3>The Einstein Summation Convention<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=518s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">The Einstein summation convention<\/span><\/strong><\/a> allows us to simplify the representation of vectors in general and the dot product in particular. Observing the two expressions above, we see that the subscript <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i<\/span><\/span> appears both in the vector coefficient and in the basis vector; for Einstein, the presence of repeated indices is enough to assume the existence of the summation in the expression, so we can write:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=  x_i\\hat{e}_i<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}= y_i\\hat{e}_i <\/span>\n<p>Using this notation convention, the dot product is expressed as follows:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = x_i\\hat{e}_i \\cdot y_i\\hat{e}_i = x_iy_i \\underbrace{(\\hat{e}_i \\cdot \\hat{e}_i)}_{=1} = x_iy_i  <\/span>\n<p>In this last equality, it has been assumed that we are working with the canonical basis.<\/p>\n<h3>Other Notations for the Dot Product<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=825s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">The notation for vectors and their operations is not always the same in all contexts,<\/span><\/strong><\/a> the one I have used in the first paragraphs of this entry is the most commonly seen when working in calculus. When working in linear algebra, sometimes a distinction is made between vectors and covectors:<\/p>\n<p>When we talk about vectors, we refer to what is known as a \u00abcolumn vector,\u00bb which is represented in matrix form as:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha^i = \\left( \\begin{array}{c}\\alpha_1 \\\\ \\vdots \\\\ \\alpha_n \\end{array} \\right)  <\/span>\n<p>Whereas when we talk about covectors, we refer to what is called a \u00abrow vector,\u00bb which is represented in matrix form as:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_i = \\left( \\beta_1 \\; \\cdots \\; \\beta_n  \\right)  <\/span>\n<p>Thus, the dot product of two vectors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots,y_n)<\/span><\/span> is interpreted as the matrix product of the \u00abcovector\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> with the vector <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^i,<\/span><\/span> yielding the following real number:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( x_1 \\; \\cdots \\; x_n  \\right) \\left( \\begin{array}{c}y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right)  = x_iy^i  <\/span>\n<p>Observe that in this last equality, the Einstein summation convention appears again, as the repeated indices indicate that the final result is a sum.<\/p>\n<p>The notation that distinguishes vectors and covectors using subscripts and superscripts is known as \u00abcovariant notation\u00bb or \u00abtensor notation\u00bb and is widely used in studying special and general relativity. This notation also has the advantage of facilitating work with tensors, a concept that generalizes the ideas we have just reviewed and that we will examine in more detail on another occasion. In other disciplines, such as quantum mechanics, the Bra-Ket notation is preferred, where:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; x \\right| =\\left( x_1 \\; \\cdots \\; x_n  \\right) \\\\ \\\\ \\left|y\\right&gt; = \\left( \\begin{array}{c}y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right)\n\n <\/span>\n<p>Thus, the dot product is represented as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt;x|y\\right&gt;.<\/span><\/span><\/p>\n<h3>Properties of the Dot Product<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=1083s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">From the definition of the dot product, we can derive a whole series of properties<\/span><\/strong><\/a> that will be highly relevant in the future.<\/p>\n<p>If we use the dot product to define the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}(\\vec{x})=\\vec{\\omega} \\cdot \\vec{x} = \\omega_i x^i,<\/span><\/span> we see that the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> defined in this way possesses all the properties of linear functions, as it is straightforward to prove that<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\tilde{\\omega}(\\alpha \\vec{x} + \\beta\\vec{y}) = \\alpha \\tilde{\\omega}(\\vec{x}) + \\beta\\tilde{\\omega}(\\vec{y}) \\end{array}<\/span>\n<p>For this reason, objects such as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> that are defined using the dot product are called <strong>linear functionals.<\/strong> As we already know, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> is a vector belonging to the <strong>vector space<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n,<\/span><\/span> and as we will see in other circumstances, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> is an object in the <strong>dual space<\/strong> of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span><\/p>\n<p>From this, it follows that there is a close relationship between the dot product and linear functions; in fact, a statement that summarizes all the important properties of the dot product is: <em><strong>\u00abThe dot product is a bilinear, symmetric, positive, and non-degenerate form.\u00bb<\/strong><\/em> Let&#8217;s examine what each part of this statement means:<\/p>\n<p>When we say that <strong>the dot product is a bilinear form,<\/strong> we mean that if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> are vectors in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\in \\mathbb{R},<\/span><\/span> then the following two equalities hold:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\vec{x}\\cdot(\\alpha \\vec{y} + \\beta\\vec{z}) = \\alpha (\\vec{x}\\cdot\\vec{y}) + \\beta(\\vec{x}\\cdot\\vec{z}) \\\\ \\\\ (\\alpha \\vec{x} + \\beta\\vec{y})\\cdot\\vec{z} = \\alpha (\\vec{x} \\cdot \\vec{z}) + \\beta(\\vec{y}\\cdot\\vec{z}) \\end{array}<\/span>\n<p>The dot product <strong>is symmetric<\/strong> because:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall(\\vec{x},\\vec{y}\\in\\mathbb{R}^n)(\\vec{x}\\cdot\\vec{y} = \\vec{y}\\cdot\\vec{x})<\/span>\n<p>It is <strong>positive definite<\/strong> because:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\vec{x}\\in\\mathbb{R}^n)(\\vec{x}\\cdot\\vec{x} \\geq 0)<\/span>\n<p>And finally, it is <strong>non-degenerate<\/strong> because:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{x} = 0 \\leftrightarrow \\vec{x}=\\vec{0}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/vTFqDBEyU4Y\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>The Norm and the Euclidean Distance<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=174s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">A norm is a way to measure the magnitude of a vector,<\/span><\/strong><\/a> when a vector space has a norm, it is called a <strong>Normed Vector Space.<\/strong> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> then the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm( . )<\/span><\/span> is a norm if it satisfies the following properties: <\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x})\\geq 0<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x}) = 0 \\leftrightarrow \\vec{x}=\\vec{0}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\lambda\\vec{x}) = |\\lambda| Norm(\\vec{x})<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x} + \\vec{y}) \\leq Norm(\\vec{x}) + Norm(\\vec{y})<\/span><\/span><\/li>\n<\/ol>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=350s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">An important aspect of the dot product<\/span><\/strong><\/a> is that it is particularly useful for defining a mathematical concept of distance that intuitively aligns with our natural understanding of distances between two points. For each <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span>, its <strong>Euclidean Norm,<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|<\/span><\/span> is defined by the equation:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\| = \\sqrt{\\vec{x}\\cdot\\vec{x}}<\/span>\n<p>From this, we say that <strong>the Euclidean norm is the norm induced by the dot product.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=846s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">A distance, or metric,<\/span><\/strong><\/a> is a function that tells us about \u00abthe separation between two elements of a set.\u00bb If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}, \\vec{y}, \\vec{z}\\in\\mathbb{R}^n<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span>, then the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist( . )<\/span><\/span> is a distance if it satisfies the following properties:<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{y})=0 \\leftrightarrow \\vec{x}=\\vec{y}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{y})=Dist(\\vec{y},\\vec{x})\\geq 0<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{z})\\leq Dist(\\vec{x},\\vec{y}) + Dist(\\vec{y},\\vec{z})<\/span><\/span><\/li>\n<\/ol>\n<p>The last expression is known as the <strong>Triangle Inequality,<\/strong> and if it were not satisfied, the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(.)<\/span><\/span> would be what is called a \u00abpseudo-distance\u00bb or \u00abpseudo-metric.\u00bb A Vector Space equipped with a distance is known as a <strong>Metric Space.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1013s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">From the Euclidean Norm,<\/span><\/strong><\/a> we define the <strong>Euclidean Distance<\/strong> between two vectors. If we have two vectors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span> then the Euclidean distance between these two vectors, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y})<\/span><\/span>, is given by:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y}) = \\|\\vec{x} - \\vec{y}\\|<\/span>\n<p>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots, y_n),<\/span><\/span> then it is easy to prove from the properties of the dot product and the norm that:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y}) = \\sqrt{\\displaystyle \\sum_{i=1}^n (x_i - y_i)^2}<\/span>\n<p>If we equip the vector space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> with the Euclidean distance, what we obtain is a <strong>Euclidean Space.<\/strong><\/p>\n<p>From this, we say that <strong>the metric of Euclidean space is the metric induced by the Euclidean norm.<\/strong><\/p>\n<h3>Properties of the Euclidean Norm<\/h3>\n<p><\/strong> Since our study focuses specifically on Euclidean Space, it will be useful to review the properties of the Euclidean norm.<\/p>\n<h4>Cauchy-Schwarz Inequality<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> then the following property holds:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|\\vec{x}\\cdot\\vec{y}|\\leq \\|\\vec{x}\\|\\|\\vec{y}\\|<\/span>\n<p>PROOF:<\/p>\n<p>Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda = (\\vec{x}\\cdot\\vec{y})\/\\|\\vec{y}\\|^2,<\/span><\/span> then we have:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} 0\\leq \\|\\vec{x} - \\lambda \\vec{y}\\|^2 &amp;= (\\vec{x} - \\lambda\\vec{y}) \\cdot (\\vec{x} - \\lambda\\vec{y}) \\\\ \\\\\n\n\\displaystyle &amp;= \\vec{x}\\cdot\\vec{x} - \\lambda\\vec{x}\\cdot\\vec{y} + \\lambda\\vec{y}\\cdot\\vec{x} + \\lambda^2(\\vec{y}\\cdot\\vec{y})\\\\ \\\\ &amp;= \\|\\vec{x}\\|^2 - 2\\lambda(\\vec{x}\\cdot\\vec{y}) + \\lambda^2 \\|\\vec{y}\\|^2 \\\\ \\\\\n\n\\displaystyle &amp;= \\|\\vec{x}\\|^2 - 2\\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|^2}\\right)(\\vec{x}\\cdot\\vec{y}) + \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{{\\|\\vec{y}\\|^2}}\\right)^2 {\\|\\vec{y}\\|^2}\\\\ \\\\\n\n\\displaystyle &amp;= \\|\\vec{x}\\|^2 - 2\\left(\\frac{(\\vec{x}\\cdot\\vec{y})^2}{\\|\\vec{y}\\|^2}\\right) + \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2}\\\\ \\\\ &amp;= \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} \\end{array}<\/span>\n<p>Thus, we can state: <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 0 \\leq \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} <\/span>\n<p>And therefore: <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\vec{x}\\cdot\\vec{y}\\right)^2 \\leq \\|\\vec{x}\\|^2 \\|\\vec{y}\\|^2 <\/span>\n<p>Finally, taking square roots, we arrive at what we wanted to prove: <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\|<\/span> \u2b1b<\/p>\n<h4>Triangle Inequality<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=2065s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> these vectors satisfy the relation:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} + \\vec{y}\\| \\leq \\|\\vec{x}\\| + \\|\\vec{y}\\|<\/span>\n<p>PROOF:<\/p>\n<p>First, let&#8217;s note that:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x} + \\vec{y}\\|^2 &amp;= (\\vec{x} + \\vec{y})\\cdot(\\vec{x} + \\vec{y}) \\\\ \\\\ &amp;=\\|\\vec{x}\\|^2 + 2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 \\end{array}<\/span>\n<p>Since the following inequalities hold:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}\\leq |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\|\\vec{y}\\|<\/span>\n<p>We can write the following:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\|\\vec{x} + \\vec{y}\\|^2 &amp;\\leq  \\|\\vec{x}\\|^2 + 2\\|\\vec{x}\\|\\|\\vec{y}\\| + \\|\\vec{y}\\|^2 \\\\ \\\\  &amp;\\leq  \\left(\\|\\vec{x}\\|  + \\|\\vec{y}\\| \\right)^2\n\n\\end{array}<\/span>\n<p>Finally, taking square roots, we arrive at what we wanted to prove:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} + \\vec{y}\\|\\leq  \\|\\vec{x}\\|  + \\|\\vec{y}\\|<\/span> \u2b1b <\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Conclusion<\/h2>\n<p>Throughout this class, we have explored the fundamental properties of Euclidean space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>, addressing its algebraic and metric structures. We began by defining its basic operations, such as vector addition and scalar multiplication, thereby establishing its nature as a vector space. Then, we delved into the concept of the dot product and its relevance to the geometry of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>, highlighting its matrix interpretation and its relationship with linear functions.<\/p>\n<p>Subsequently, we analyzed the Euclidean norm and the distance it induces, emphasizing how these tools allow us to quantify lengths and distances in this space. Additionally, we reviewed fundamental properties such as the Cauchy-Schwarz inequality:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\| <\/span>\n<p>and the triangle inequality:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\|\\vec{x} + \\vec{y}\\|\\leq  \\|\\vec{x}\\|  + \\|\\vec{y}\\| <\/span>\n<p>which are key to the development of more advanced theories in analysis and geometry.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Euclidean Space In this class, we explore the Euclidean space , its algebraic structure, and metric properties. You will learn about vector operations, the dot product, the norm, and the Euclidean distance, essential concepts in geometry and analysis. With clear explanations and intuitive examples, this material will help you understand how space is mathematically [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32541,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":10,"footnotes":""},"categories":[791,567,1116],"tags":[],"class_list":["post-32543","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-linear-algebra","category-mathematics","category-multivariable-calculus"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Euclidean Space Rn - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubre las propiedades del Espacio Euclidiano Rn, su estructura algebraica, el producto escalar, la norma y la distancia euclidiana en m\u00faltiples dimensiones.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/the-euclidean-space-rn\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Euclidean Space Rn\" \/>\n<meta property=\"og:description\" content=\"Descubre las propiedades del Espacio Euclidiano Rn, su estructura algebraica, el producto escalar, la norma y la distancia euclidiana en m\u00faltiples dimensiones.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/the-euclidean-space-rn\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-08T13:00:42+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-03-10T01:31:47+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectores-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The Euclidean Space Rn\" \/>\n<meta name=\"twitter:description\" content=\"Descubre las propiedades del Espacio Euclidiano Rn, su estructura algebraica, el producto escalar, la norma y la distancia euclidiana en m\u00faltiples dimensiones.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectores.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Euclidean Space Rn\",\"datePublished\":\"2022-03-08T13:00:42+00:00\",\"dateModified\":\"2025-03-10T01:31:47+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/\"},\"wordCount\":2856,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/vectores.jpg\",\"articleSection\":[\"Linear Algebra\",\"Mathematics\",\"Multivariable Calculus\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-space-rn\\\/\",\"name\":\"The Euclidean Space Rn - 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