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"file": "Maximally Linearly Independent.md", + "file": "Orthogonal Projections.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Maximally Linearly Independent" + "title": "Orthogonal Projections" } }, { @@ -177,12 +191,12 @@ "state": { "type": "markdown", "state": { - "file": "Minimal Generating.md", + "file": "Cosine Similarity.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Minimal Generating" + "title": "Cosine Similarity" } }, { @@ -191,12 +205,12 @@ "state": { "type": "markdown", "state": { - "file": "Span.md", + "file": "Gram-Schmidt orthogonalization process.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Span" + "title": "Gram-Schmidt orthogonalization process" } }, { @@ -205,12 +219,54 @@ "state": { "type": "markdown", "state": { - "file": "Non-Trivial.md", + "file": "Inner Products.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Non-Trivial" + "title": "Inner Products" + } + }, + { + "id": "9639dd8dbd93ff0c", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Inner Products.md", + "mode": "preview", + "source": false + }, + "icon": "lucide-file", + "title": "Inner Products" + } + }, + { + "id": "f35edcb5f3f1a4ed", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Matrix Multiplication.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Matrix Multiplication" + } + }, + { + "id": "bc3c73f8a18068c0", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Matricies.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Matricies" } }, { @@ -219,16 +275,78 @@ "state": { "type": "markdown", "state": { - "file": "Vector Set.md", + "file": "Bi-linearity.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Vector Set" + "title": "Bi-linearity" + } + }, + { + "id": "d5d74d65f58ce956", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Linear Transformation.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Linear Transformation" + } + }, + { + "id": "ef6ba7403e322484", + "type": "leaf", + "state": { + "type": "graph", + "state": {}, + "icon": "lucide-git-fork", + "title": "Graph view" + } + }, + { + "id": "9c5d7c233e891204", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Surjective Transformations.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Surjective Transformations" + } + }, + { + "id": "09c013eb7afd706d", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Injective Transformations.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Injective Transformations" + } + }, + { + "id": "40d18dc580226ba0", + "type": "leaf", + "state": { + "type": "graph", + "state": {}, + "icon": "lucide-git-fork", + "title": "Graph view" } } ], - "currentTab": 10 + "currentTab": 24 } ], "direction": "vertical" @@ -389,32 +507,37 @@ "github-sync:Sync with Remote": false } }, - "active": "aa4d19e672f564d3", + "active": "40d18dc580226ba0", "lastOpenFiles": [ - "Null Vector.md", - "P-Norms.md", + "Bijective Transformations.md", + "Injective Transformations.md", + "Surjective Transformations.md", "Span.md", - "Pasted image 20260211175349.png", - "Orthogonal Projections.md", - "Inner Products.md", - "Basis.md", + "Subspaces.md", + "Matrix Multiplication.md", + "Linear Transformation.md", + "Transformation.md", "Vectors.md", - "Generating Set.md", - "Supremum Norm.md", - "Normed Spaces.md", - "Norm.md", - "Magnitude.md", + "Pasted image 20260420174553.png", + "Matrices.md", + "Bi-linearity.md", + "Basis.md", + "Vector Set.md", + "Inner Products.md", + "Inner Product Spaces.md", + "Gram-Schmidt orthogonalization process.md", + "Orthogonal Projections.md", + "Linear Dependency.md", + "Cosine Similarity.md", "Cardinality.md", "Dimensionality.md", - "Vector Set.md", - "Maximally Linearly Independent.md", - "Vector Spaces.md", - "Subspaces.md", - "Minimal Generating.md", - "Linear Dependency.md", - "Linear Combinations.md", + "Generating Set.md", "Non-Trivial.md", - "Sets.md", - "Welcome.md" + "Orthogonality.md", + "Untitled.canvas", + "Untitled.base", + "Minimal Generating.md", + "P-Norms.md", + "Pasted image 20260211175349.png" ] } \ No newline at end of file diff --git a/AI Linear Algebra/Bi-linearity.md b/AI Linear Algebra/Bi-linearity.md new file mode 100644 index 0000000..62ce669 --- /dev/null +++ b/AI Linear Algebra/Bi-linearity.md @@ -0,0 +1,3 @@ +A property of 2D functions that says both axes are linear no matter what. + +%%related to [[Vectors]]%% \ No newline at end of file diff --git a/AI Linear Algebra/Bijective Transformations.md b/AI Linear Algebra/Bijective Transformations.md new file mode 100644 index 0000000..a2fd298 --- /dev/null +++ b/AI Linear Algebra/Bijective Transformations.md @@ -0,0 +1 @@ +Bijective transformations are both [[Surjective Transformations|Surjective]] and [[Injective Transformations|Injective]]. \ No newline at end of file diff --git a/AI Linear Algebra/Cosine Similarity.md b/AI Linear Algebra/Cosine Similarity.md new file mode 100644 index 0000000..80498a4 --- /dev/null +++ b/AI Linear Algebra/Cosine Similarity.md @@ -0,0 +1 @@ +The cosine similarity is a way to measure the similarity of [[Vectors]]. It is defined as: $$cos(𝐱,𝐲)=\Biggl \langle\frac{𝐱}{‖𝐱‖},\frac{𝐲}{‖𝐲‖} \Biggr \rangle$$ \ No newline at end of file diff --git a/AI Linear Algebra/Gram-Schmidt orthogonalization process.md b/AI Linear Algebra/Gram-Schmidt orthogonalization process.md new file mode 100644 index 0000000..8c764c3 --- /dev/null +++ b/AI Linear Algebra/Gram-Schmidt orthogonalization process.md @@ -0,0 +1 @@ +A process that takes a [[Linear Dependency|Linearly Independent]] [[Basis]] and creates an [[Orthogonality|Orthogonal]] [[Basis]] that produces the same [[Vector Spaces|Vector Space]] as the original [[Basis]]. \ No newline at end of file diff --git a/AI Linear Algebra/Injective Transformations.md b/AI Linear Algebra/Injective Transformations.md new file mode 100644 index 0000000..98f2e6e --- /dev/null +++ b/AI Linear Algebra/Injective Transformations.md @@ -0,0 +1 @@ +Injective [[Transformation|Transformations]] are a type of transformation that distinctly maps between [[Vector Spaces]]. \ No newline at end of file diff --git a/AI Linear Algebra/Inner Product Spaces.md b/AI Linear Algebra/Inner Product Spaces.md new file mode 100644 index 0000000..c6dc1f2 --- /dev/null +++ b/AI Linear Algebra/Inner Product Spaces.md @@ -0,0 +1 @@ +Inner Product Spaces are [[Vector Spaces]] that have an [[Inner Products|Inner Product]] \ No newline at end of file diff --git a/AI Linear Algebra/Inner Products.md b/AI Linear Algebra/Inner Products.md index bd2c717..b5a5a92 100644 --- a/AI Linear Algebra/Inner Products.md +++ b/AI Linear Algebra/Inner Products.md @@ -1,3 +1,4 @@ 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. \ No newline at end of file +inner products have the unique property that if two vectors are orthogonal from each other that their inner product is 0. +(This process is also know as a dot product) \ No newline at end of file diff --git a/AI Linear Algebra/Linear Transformation.md b/AI Linear Algebra/Linear Transformation.md new file mode 100644 index 0000000..d5953f8 --- /dev/null +++ b/AI Linear Algebra/Linear Transformation.md @@ -0,0 +1 @@ +A [[Transformation]] is linear if the transformation can be expressed like this: $𝑓(𝑎𝐱 + 𝑏𝐲) = 𝑎𝑓 (𝐱) + 𝑏𝑓 (𝐲)$ \ No newline at end of file diff --git a/AI Linear Algebra/Matrices.md b/AI Linear Algebra/Matrices.md new file mode 100644 index 0000000..2dff5ff --- /dev/null +++ b/AI Linear Algebra/Matrices.md @@ -0,0 +1 @@ +Matrices are a multidimensional array of numbers that can be operated on. diff --git a/AI Linear Algebra/Matrix Multiplication.md b/AI Linear Algebra/Matrix Multiplication.md new file mode 100644 index 0000000..3282c12 --- /dev/null +++ b/AI Linear Algebra/Matrix Multiplication.md @@ -0,0 +1,2 @@ +Matrix multiplication is an operation on two [[Matrices]] where each row of the first matrix is scaled and summed up by the second. +![[Pasted image 20260420174553.png]] \ No newline at end of file diff --git a/AI Linear Algebra/Orthogonality.md b/AI Linear Algebra/Orthogonality.md new file mode 100644 index 0000000..2530d15 --- /dev/null +++ b/AI Linear Algebra/Orthogonality.md @@ -0,0 +1 @@ +A property of a [[Basis]] that says all elements are orthogonal. \ No newline at end of file diff --git a/AI Linear Algebra/Pasted image 20260420174553.png b/AI Linear Algebra/Pasted image 20260420174553.png new file mode 100644 index 0000000..f5e664d Binary files /dev/null and b/AI Linear Algebra/Pasted image 20260420174553.png differ diff --git a/AI Linear Algebra/Surjective Transformations.md b/AI Linear Algebra/Surjective Transformations.md new file mode 100644 index 0000000..1f9e9a5 --- /dev/null +++ b/AI Linear Algebra/Surjective Transformations.md @@ -0,0 +1 @@ +Surjective [[Transformation|Transformations]] are transformations that map one [[Vector Spaces|Vector Space]] onto another but are lossy and can include repeats. \ No newline at end of file diff --git a/AI Linear Algebra/Transformation.md b/AI Linear Algebra/Transformation.md new file mode 100644 index 0000000..cca3feb --- /dev/null +++ b/AI Linear Algebra/Transformation.md @@ -0,0 +1 @@ +An operation on [[Vectors]] or [[Matrices]] \ No newline at end of file diff --git a/AI Linear Algebra/Vectors.md b/AI Linear Algebra/Vectors.md index e1a01af..77e60f6 100644 --- a/AI Linear Algebra/Vectors.md +++ b/AI Linear Algebra/Vectors.md @@ -1,4 +1,6 @@ Vectors are a finite list of values -They can also be contained in [[Vector Set]] \ No newline at end of file +They can also be contained in [[Vector Set]] + +(Can also be thought of as a 1 dimensional [[Matrices|Matrix]]) \ No newline at end of file