From 7bb5d5d762d8725238b05e66601890281a6d910a Mon Sep 17 00:00:00 2001 From: Rivulet Date: Wed, 11 Feb 2026 18:13:10 -0800 Subject: [PATCH] drainagesystem 2026-2-11:18:13:10 --- AI Linear Algebra/.obsidian/workspace.json | 39 +++++++++++++----- AI Linear Algebra/Inner Products.md | 3 ++ AI Linear Algebra/Orthogonal Projections.md | 7 ++++ AI Linear Algebra/P-Norms.md | 2 +- .../Pasted image 20260211175349.png | Bin 0 -> 29290 bytes 5 files changed, 39 insertions(+), 12 deletions(-) create mode 100644 AI Linear Algebra/Inner Products.md create mode 100644 AI Linear Algebra/Orthogonal Projections.md create mode 100644 AI Linear Algebra/Pasted image 20260211175349.png diff --git a/AI Linear Algebra/.obsidian/workspace.json b/AI Linear Algebra/.obsidian/workspace.json index 61347ee..accd311 100644 --- a/AI Linear Algebra/.obsidian/workspace.json +++ b/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" } }, { @@ -214,7 +228,7 @@ } } ], - "currentTab": 7 + "currentTab": 10 } ], "direction": "vertical" @@ -375,23 +389,26 @@ "github-sync:Sync with Remote": false } }, - "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", diff --git a/AI Linear Algebra/Inner Products.md b/AI Linear Algebra/Inner Products.md new file mode 100644 index 0000000..bd2c717 --- /dev/null +++ b/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: $āŸØš±, š²āŸ© =āˆ‘^š‘›_{š‘–=1}š‘„_š‘–š‘¦_š‘–$ +inner products have the unique property that if two vectors are orthogonal from each other that their inner product is 0. \ No newline at end of file diff --git a/AI Linear Algebra/Orthogonal Projections.md b/AI Linear Algebra/Orthogonal Projections.md new file mode 100644 index 0000000..cf51a75 --- /dev/null +++ b/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: $⟨x āˆ’ š‘y, y⟩ = 0$ then to solve for c like this: $$⟨x āˆ’ š‘y, y⟩ = 0 +$$$$⟨x, y⟩-š‘āŸØy, y⟩ = 0$$ $$⟨x, y⟩ = c⟨y, y⟩$$ $$\frac{⟨x, y⟩}{⟨y, y⟩} = š‘$$ +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 $c=\frac{⟨x, y⟩}{⟨y, y⟩}$. \ No newline at end of file diff --git a/AI Linear Algebra/P-Norms.md b/AI Linear Algebra/P-Norms.md index ca1f550..7ace358 100644 --- a/AI Linear Algebra/P-Norms.md +++ b/AI Linear Algebra/P-Norms.md @@ -1,2 +1,2 @@ P-norms are a type of [[Norm]] that follows this formula: -$$ā€–š±ā€–_š‘ = \biggl(āˆ‘^š‘›_{š‘–=1}|š‘„š‘–|^š‘\biggr)^{1/š‘},š± = (š‘„_1, … , š‘„_š‘›)$$ \ No newline at end of file +$$ā€–š±ā€–_š‘ = \biggl(āˆ‘^š‘›_{š‘–=1}|š‘„š‘–|^š‘\biggr)^{1/š‘},š± = (š‘„_1, … , š‘„_š‘›)$$ diff --git a/AI Linear Algebra/Pasted image 20260211175349.png b/AI Linear Algebra/Pasted image 20260211175349.png new file mode 100644 index 0000000000000000000000000000000000000000..cbf8b3c6ffac634d620c1d437225b89121e77686 GIT binary patch literal 29290 zcmeGEXHb;Q6F!VChzSHy#DF9dCX{219-K z&aL|x3A_<^uPt&LwMq=1!pGxEz@Hl<63;Pvz^2^OrdSDPP{nP5Q82Hr~%4dFUN=46SkF zp4(#dqjekX4+XQe-t`Lhrg1iS7W_w%b&ZVl2iC-6n16qw5GH3O{Sy`=hoK<-iFV`= zmh=YFMcd4lXW@X9uX?|MwrOdwO#6O}%0p 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