drainagesystem 2026-2-11:18:13:10

AI Linear Algebra/.obsidian/workspace.json

@@ -107,12 +107,12 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "Supremum Norm.md",
+                "file": "P-Norms.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "Supremum Norm"
+              "title": "P-Norms"
             }
           },
           {
@@ -135,12 +135,26 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "Basis.md",
+                "file": "Inner Products.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "Basis"
+              "title": "Inner Products"
+            }
+          },
+          {
+            "id": "aa4d19e672f564d3",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "Orthogonal Projections.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Orthogonal Projections"
             }
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@@ -214,7 +228,7 @@
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-        "currentTab": 7
+        "currentTab": 10
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     "direction": "vertical"
@@ -375,23 +389,26 @@
       "github-sync:Sync with Remote": false
     }
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-  "active": "2c3f77128f00e180",
+  "active": "aa4d19e672f564d3",
   "lastOpenFiles": [
+    "Null Vector.md",
     "P-Norms.md",
+    "Span.md",
+    "Pasted image 20260211175349.png",
+    "Orthogonal Projections.md",
+    "Inner Products.md",
+    "Basis.md",
+    "Vectors.md",
+    "Generating Set.md",
     "Supremum Norm.md",
     "Normed Spaces.md",
     "Norm.md",
     "Magnitude.md",
-    "Null Vector.md",
     "Cardinality.md",
-    "Basis.md",
-    "Span.md",
-    "Vectors.md",
     "Dimensionality.md",
     "Vector Set.md",
     "Maximally Linearly Independent.md",
     "Vector Spaces.md",
-    "Generating Set.md",
     "Subspaces.md",
     "Minimal Generating.md",
     "Linear Dependency.md",

AI Linear Algebra/Inner Products.md

@@ -0,0 +1,3 @@
+The inner product is a operation that takes in two [[Vectors]] and outputs a number
+where the inner product can be defined like this: $⟨𝐱, 𝐲⟩ =∑^𝑛_{𝑖=1}𝑥_𝑖𝑦_𝑖$
+inner products have the unique property that if two vectors are orthogonal from each other that their inner product is 0.

AI Linear Algebra/Orthogonal Projections.md

@@ -0,0 +1,7 @@
+Orthogonal Projections are a type of projection that maps one vector onto another:
+![[Pasted image 20260211175349.png]]
+In this image the result for the projection of $x$ onto $y$ is $x_p$
+in order to derive this we can define $x_p = cy$ where $c$ is a mysterious scalar as $x_p$ is always in the same direction as $Y$ and we can define $x_o = x-x_p = x-cy$ and because $x_p$ is orthogonal we can define it as an Inner Product %%[[Inner Products]]%% like this: $⟨x − 𝑐y, y⟩ = 0$ then to solve for c like this: $$⟨x − 𝑐y, y⟩ = 0
+$$$$⟨x, y⟩-𝑐⟨y, y⟩ = 0$$ $$⟨x, y⟩ = c⟨y, y⟩$$ $$\frac{⟨x, y⟩}{⟨y, y⟩} = 𝑐$$
+so after all that we get $c=\frac{⟨x, y⟩}{⟨y, y⟩}$ from this we can derive $x_p$ by doing this: $$x_p = \frac{⟨x, y⟩}{⟨y, y⟩}y$$
+because $x_p = cy$ and $c=\frac{⟨x, y⟩}{⟨y, y⟩}$.