minor things

Author: LearningAlgorithms

_includes/js-selector.html

@@ -18,8 +18,23 @@
 
 {% if page.math %}
   <!-- MathJax -->
-  <script defer src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
-  <script defer src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+  <script type="text/x-mathjax-config">
+    MathJax.Hub.Config({
+      tex2jax: {
+        skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code'],
+        inlineMath: [['$','$'], ['\\(','\\)']],
+        processEscapes: true
+      },
+      TeX: {
+        equationNumbers: {
+          autoNumber: "AMS"
+        }
+      }
+    });
+  </script>
+  <script type="text/javascript" async
+    src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS_CHTML">
+  </script>
 {% endif %}
 
 {% if jekyll.environment == 'production' %}

…hine-learning/2020-11-03-introduction.md …hine-learning/2020-11-05-introduction.md_posts/machine-learning/2020-11-03-introduction.md renamed to _posts/machine-learning/2020-11-05-introduction.md _posts/machine-learning/2020-11-03-introduction.md renamed to _posts/machine-learning/2020-11-05-introduction.md

@@ -15,32 +15,32 @@ Prepostscript (PPS):  This post will be updated when more knowledge of linear al
 
 ## Introduction:
 
-You may have noticed that **Linear Algebra** is a term composed by _Linear_ + _Algebra_. So let's start by defining what is each of the two terms. An _algebra_ it's the union of a **set of objects** and a **set of rules** that applies to that objects. Etymologically the word _Algebra_ comes from the title of the book  [_Ilm al-jabr wa l-muqābala_](https://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing), '_The Compendious Book on Calculation by Completion and Balancing_', by the Persian mathematician [al-Khwarizmi](https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi), also, the word _Algorithm_ derivates from his name, so you can imagine that he was an influential mathematician. The word _Linear_  refers to that what we are traying to solve using those set of objects and rules are **Linear systems**. Systems of equations where those equations are of first order, the maximum degree of the variables is equal to one, so in $$\mathbb R^2$$ (in the plane) we have straight lines, in $$ \mathbb R ^3$$​  (in a space) we have a plane.
+You may have noticed that **Linear Algebra** is a term composed by _Linear_ + _Algebra_. So let's start by defining what is each of the two terms. An _algebra_ it's the union of a **set of objects** and a **set of rules** that applies to that objects. Etymologically the word _Algebra_ comes from the title of the book  [_Ilm al-jabr wa l-muqābala_](https://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing), '_The Compendious Book on Calculation by Completion and Balancing_', by the Persian mathematician [al-Khwarizmi](https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi), also, the word _Algorithm_ derivates from his name, so you can imagine that he was an influential mathematician. The word _Linear_  refers to that what we are traying to solve using those set of objects and rules are **Linear systems**. Systems of equations where those equations are of first order, the maximum degree of the variables is equal to one, so in $\mathbb R^2$ (in the plane) we have straight lines, in $\mathbb R ^3$​  (in a space) we have a plane.
 
-Those objects in our case, in ML, will be **vectors**, elements of $$\mathbb R ^n$$ (tuples of $$n$$ real numbers). For example, $$v$$ can be a vector of $$\mathbb R ^2$$ ($$ v \in \mathbb R ^2$$) with coordinates $$v_1 = 42$$ and $$v_2 = 73$$: 
+Those objects in our case, in ML, will be **vectors**, elements of $\mathbb R ^n$ (tuples of $n$ real numbers). For example, $v$ can be a vector of $\mathbb R ^2$ ($ v \in \mathbb R ^2$) with coordinates $v_1 = 42$ and $v_2 = 73$: 
 
 $$ v  =  \begin{bmatrix} 42 \cr 73 \end{bmatrix}\in \mathbb R ^2 $$
 
-We can represent a vector of $$w \in \mathbb R^n$$:
+We can represent a vector of $w \in \mathbb R^n$:
 
 $$ w  =  \begin{bmatrix} w_1 \cr w_2 \cr . \cr . \cr . \cr w_n \end{bmatrix}\in \mathbb R ^n $$
 
 And the **rules** or _axioms_ that apply to those vectors are the following:
 
-Being $$u, v, w$$ vectors, and $$a, b, 0$$ and $$1$$ _scalars_ (elements of $$\mathbb R$$), then:
+Being $u, v, w$ vectors, and $a, b, 0$ and $1$ _scalars_ (elements of $\mathbb R$), then:
 
-- (A1) $$ u + (v + w) = (u + v) + w$$
-- (A2) $$u + v = v + u$$
-- (A3) Exists a vector, called the zero vector, such that $$v +0 = v$$
-- (A4) For every vector $$v$$ exists a vector $$-v$$ called the inverse
-- (A5) $$a(u+v) = au + av$$
-- (A6) $$(a + b) v = av + bv$$
-- (A7) $$ a(bv)  = (ab)v$$
-- (A8) Exist a scalar called the identity, and denoted as $$1$$, such as $$1*v = v$$
+- (A1) $ u + (v + w) = (u + v) + w$
+- (A2) $u + v = v + u$
+- (A3) Exists a vector, called the zero vector, such that $v +0 = v$
+- (A4) For every vector $v$ exists a vector $-v$ called the inverse
+- (A5) $a(u+v) = au + av$
+- (A6) $(a + b) v = av + bv$
+- (A7) $ a(bv)  = (ab)v$
+- (A8) Exist a scalar called the identity, and denoted as $1$, such as $1*v = v$
 
 Mathematically those rules are important, since they are defining how vectors interact with themselves and with scalars. 
 
-Now, specifically, we can define those operations on vectors of $$\mathbb R^n$$ .  For example, if $$u, v \in \mathbb R^2$$ and $$a \in \mathbb R$$, then:
+Now, specifically, we can define those operations on vectors of $\mathbb R^n$ .  For example, if $u, v \in \mathbb R^2$ and $a \in \mathbb R$, then:
 
 $$ u + v = \begin{bmatrix} u_1 \cr u_2 \end{bmatrix} + \begin{bmatrix} v_1 \cr v_2 \end{bmatrix} = \begin{bmatrix} u_1 + v_1 \cr u_2+ v_2 \end{bmatrix} $$