update linear algebra

.vscode/settings.json

@@ -1,5 +1,12 @@
 {
-  "editor.formatOnSave": true,
   "explorer.excludeGitIgnore": true,
   "latex-workshop.latex.autoBuild.run": "never",
+  "filewatcher.commands": [
+    {
+      "match":"trees/",
+      "isAsync": true,
+      "cmd": "cd ${currentWorkspace} && make",
+      "event": "onFolderChange"
+    }
+  ]
 }

theme/forest.xsl

@@ -172,7 +172,7 @@
             <nav class="nav">
               <div class="logo">
                 <a href="index.xml" title="Home">
-                  <xsl:text>« Home</xsl:text>
+                  <xsl:text>Home</xsl:text>
                 </a>
               </div>
             </nav>

trees/def.tree

@@ -13,4 +13,16 @@
 \transclude{def-000C}
 \transclude{def-000D}
 \transclude{def-000E}
-\transclude{def-000F}
+\transclude{def-000F}
+\transclude{def-000G}
+\transclude{def-000H}
+\transclude{def-000I}
+\transclude{def-000J}
+\transclude{def-000K}
+\transclude{def-000L}
+\transclude{def-000M}
+\transclude{def-000N}
+\transclude{def-000O}
+\transclude{def-000P}
+\transclude{def-000Q}
+\transclude{def-000R}

trees/def/def-0002.tree

@@ -1,5 +1,5 @@
 \title{Abel Group}
 \taxon{Definition}
 \p{
-If in a group #{G}, #{\forall a,b\in G, a\circ b=b\circ a} then #{G} is called an \strong{abelian (commutative) group}.
+If in a [group](def-0001) #{G}, #{\forall a,b\in G, a\circ b=b\circ a} then #{G} is called an \strong{abelian (commutative) group}.
 }

trees/def/def-0006.tree

@@ -1,6 +1,6 @@
 \title{Field}
 \taxon{Definition}
 \p{
-A field is a set #{F} together with two binary operations #{+} and #{\times} on #{F} st #{(F,+)} is an abelian group (identity is #{0}) and #{(F-\{0\},\times)} is also an abelian group such that
+A field is a set #{F} together with two binary operations #{+} and #{\times} on #{F} st #{(F,+)} is an [abelian group](def-0002) (identity is #{0}) and #{(F-\{0\},\times)} is also an abelian group such that
 #{a\times(b+c)=a\times b+a\times c,\forall a,b,c\in F}
 }

trees/def/def-000G.tree

@@ -0,0 +1,24 @@
+\title{List}
+\taxon{Definition}
+\p{
+    Let #{n} be a natural number. A \strong{list} of length #{n} is an ordered collection of #{n} elements.
+    ##{
+        (x_1, x_2, \dots, x_n)
+    }
+    Two lists are equal if and only if they have the same length and the same elements in the same order.
+}
+
+\strong{Addition in Lists}
+\p{
+    Let #{n} be a natural number. Let #{(x_1, x_2, \dots, x_n)} and #{(y_1, y_2, \dots, y_n)} be lists of length #{n}. The \strong{sum} of these lists is the list #{(x_1 + y_1, x_2 + y_2, \dots, x_n + y_n)}.
+}
+
+\strong{Additive Inverse in Lists}
+\p{
+    Let #{n} be a natural number. Let #{x=(x_1, x_2, \dots, x_n)} be a list of length #{n}. The \strong{additive inverse} of this list is the list #{-x=(-x_1, -x_2, \dots, -x_n)}.
+}
+
+\strong{Scalar Multiplication in Lists}
+\p{
+    Let #{n} be a natural number. Let #{x=(x_1, x_2, \dots, x_n)} be a list of length #{n}. Let #{c} be a real number. The \strong{scalar multiplication} of #{c} and #{x} is the list #{cx=(cx_1, cx_2, \dots, cx_n)}.
+}

trees/def/def-000H.tree

@@ -0,0 +1,29 @@
+\title{Vector Space}
+\taxon{Definition}
+\p{
+    A vector space over a [field](def-0006) #{F} is a non-empty set #{V} together with a binary operation and a binary function that satisfy the axioms listed below. 
+    In this context, the elements of #{F} are commonly called \strong{vectors}, and the elements of #{F} are called \strong{scalars}.
+    \ul{
+        \li{Commutativity: #{
+            \forall x, y \in V, x + y = y + x
+        } }
+        \li{Associativity: #{
+            \forall x, y, z \in V, (x + y) + z = x + (y + z)
+        } }
+        \li{Additive Identity: #{
+            \exists 0 \in V \text{ such that } \forall x \in V, x + 0 = x
+        } }
+        \li{Multiplicative Identity: #{
+            \forall x \in V, 1x = x
+        } }
+        \li{Additive Inverse: #{
+            \forall x \in V, \exists y \in V \text{ such that } x + y = 0
+        } }
+        \li{Distributivity: #{
+            \forall x, y \in V, \forall c, d \in F, c(x + y) = cx + cy, (c + d)x = cx + dx
+        } }
+    }
+}
+\p{
+    Elements of a vector space are called \strong{vectors} or \strong{points}.
+}