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🎬Scene234 : Functor Roughly speaking, a functor T is a function from category 𝓒 to category 𝓓. ( #math_scene227🎬) Then, through the functor T, the object from C in 𝓒 becomes T(C) in 𝓓, f∈𝘮𝘰𝘳_𝓒 (C,C') becomes T(f)∈𝘮𝘰𝘳_𝓓 (T(C),T(C')). Here is a noteworthy point: The direction of arrows on a commutative diagram. If the direction of arrow is preserved through the functor T, then we call T a 𝗰𝗼𝘃𝗮𝗿𝗶𝗮𝗻𝘁 𝗳𝘂𝗻𝗰𝘁𝗼𝗿, while we call a functor S a 𝗰𝗼𝗻𝘁𝗿𝗮𝘃𝗮𝗿𝗶𝗮𝗻𝘁 𝗳𝘂𝗻𝗰𝘁𝗼𝗿, if the arrow is reversed through the functor S. 𝗘𝘅𝗮𝗺𝗽𝗹𝗲 🟧(covariant functor) : One of the most important scenes in linear algebra is that "a linear mapping" corresponds to the "matrix". Then in the sense of the functor, let 𝓒=𝐅𝐢𝐧𝐕𝐞𝐜𝐭𝕂, (a finite-dimensional vector space), 𝓓=𝐌𝐚𝐭 (matrix). We know that there is isomorphism φ from 𝓛(V,V') to 𝕸_(n×n')(𝕂). Then a m-dimensional coordinate 𝖝=(x₁,.....xₘ) becomes a column vector 𝖝^T through the functor φ, and a linear transformation T: U→V becomes n*m-matrix through the functor φ. Then the functor φ is covariant: φ(S∘T)=φ(S)∘ φ(T) holds, which means that the composition of linear transformation corresponds to the multiplication of the matrix 🟨(contavariant functor) (See #math_scene99🎬 ) Recall what is a presheaf. Let 𝓒=𝐒𝐞𝐭, 𝓓=𝗖𝐫𝐢𝐧𝐠(collection of real-valued function), Assume that ⊆ is a morphism object in 𝓒, res_V,V' is a morphism between 𝓓, and 𝓕 is a functor (𝓕(Ω): a collection of differentiable functions on open subset Ω in ℝ.). Then 𝓕 is a typical example of presheaf and it is also a typical example of a contravariant functor. #math_scene99🎬, a visualizing example shows why the following commutative diagram holds in category 𝓓. #category #functor #covariant #contravariant #linearalgebra #abstractalgebra #algebraicgeometry #lineartransformation #ung_abstractalgebra📁 #math_scene234🎬
What is a Matrix 🤔 . . . . . . . . . . . . . . . . . . . #mathematics #mathproblems #math #scientist #physicsfun #physics #physicsmemes #mathmemes #quantumphysics #quantumcomputing #algebra #linearalgebra #physicist #ayushphy
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Quick Math Tip: 4 Math Concepts for Machine Learning. 1. Linear Algebra 2. Calculus 3. Probability and Statistics 4. Operations Research #machinelearning #datascience #mathematics #maths #artificialintelligence #optimization #linearalgebra #probability #probabilityandstatistics #statistics #calculus
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[Linear Algebra] 60. Theorem There is the theorem with its proof. #linearalgebra #basicmathematics
[Linear Algebra] 59. Trace -Defintion -Example -Theorem #linearalgebra #basicmathematics
[Linear Algebra] 58. Definition -Symmetric matrix -Skew-symmetric matrix -Hermitian matrix -Skew-Hermitian matrix #linearalgebra #basicmathematics
[Linear Algebra] 57. Definition -Transpose -Conjugate transpose -Example -Theorem #linearalgebra #basicmathematics
[Linear Algebra] 56. The Product of Submatrix -Method -Example #linearalgebra #basicmathematics
[Linear Algebra] 55. Submatrix -Definition(informal) -Example #linearalgebra #basicmathematics
[Linear Algebra] 54. Remarks There are some remarks and their example about matrix operation. #linearalgebra #basicmathematics
[Linear Algebra] 53. Theorem There is the theorem about matrix operation. #linearalgebra #basicmathematics
[Linear Algebra] 52. Theorem There is the theorem about matrix operation. #linearalgebra #basicmathematics
[Linear Algebra] 51. Product of Matrix -Definition -Example #linearalgebra #basicmathematics
[Linear Algebra] 50. Operations of Matrix -Equal -Sum -Difference -Scalar multiplication -Example #linearalgebra #basicmathematics
[Linear Algebra] 49. Definition about matrix(2) -Square matrix of order n -Diagonal element -Zero matrix -Negative matrix -Identity matrix -Upper triangular matrix -Lower triangular matrix -Strictly upper triangular matrix -Strictly lower triangular matrix -Diagonal matrix #linearalgebra #basicmathematics
[Linear Algebra] 48. Definitions about matrix -(i,j)-entry -entry of ith row -entry of jth column -A matrix with m rows and n columns -Set of all m×n matrices -Real matrix -Complex matrix -Column matrix(column vector) -Row matrix(Row vector) -Note: Definition of a field #linearalgebra #basicmathematics
[Linear Algebra] 47. Theorem There is the theorem with its proof and an example. #linearalgebra #basicmathematics
[Linear Algebra] 46. Theorem There is the theorem with its proof. #linearalgebra #basicmathematics
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Imagine re-naming det(A-xI)=0 from "characteristic equation" to "annihilator equation". . . . #mathematics #linearalgebra #undergrad #maths #proof #proofs #isomorphism #vector #vectorspace #vibe #annihilator #algebra #algebra2 #mathmajor #mathstudent
prawo bezwładności form kwadratowych Niech dimV(R) = n < ∞. Niech B i B’ będą bazami kanonicznymi formy kwadratowej F. Liczba współczynników dodatnich i ujemnych w postaci kanonicznej jest taka sama w bazach B i B’. Sygnaturą formy kwadratowej nazywamy zaś parę (p,q), gdzie p - liczba dodatnich, q - liczba ujemnych współczynników w postaci kanonicznej. #linearalgebra #rosomak
The applications of linear algebra, vectors, and matrices are of many, and the calculation of a determinant of matrix is a common mechanism to assess invertibility of matrices. Here is a quick tutorial for 2x2 matrices :) #linearalgebra #matrices #determinant #vectors #lineartransformations
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Check out the linear algebra final exam review videos on Youtube! https://www.youtube.com/@UnderstandTheMath #linearalgebra #linearalgebrareview #understandthemath The videos in this playlist explain the concepts and goes over problems that often appear in an introductory linear algebra final examination. Topics included are the following: 1: Linear Equations Solutions to a Linear System with Two Equations and Two Unknowns Coefficient and Augmented Matrices Elementary Row Operations, Echelon, and Reduced Echelon Form Gaussian Elimination and Gauss Jordan Elimination Consistent and Inconsistent Linear Systems Linear Combinations and Span 2: Matrix Algebra Addition, Subtraction, and Multiplication of Matrices Transpose and Trace of a Matrix Special Matrices - Triangular, Symmetric, and Diagonal Inverse of a 2x2 and 3x3 Matrix Solutions to a Linear System Using Matrix Inversion 3: Transformations Linear Transformations - Domain and Codomain Linear Transformations - Standard Matrix Matrix Transformations - Reflection, Projection, Rotation One-To-One and Onto Transformations 4: Determinants Determinant of a 2x2 Matrix Determinant of a 3x3 Matrix - Cofactor Expansion, Shortcut, Row Reduction Properties of Determinants Cramer’s Rule 5: Vector Spaces Linear Independence Basis Coordinate Vectors Dimension Row Space, Column Space, and Null Space Rank of a Matrix and Rank Theorem 6: Eigenvalues and Eigenvectors Introduction to Eigenvalues and Eigenvectors How to Find Eigenvalues How to Find Eigenvectors and a Basis for an Eigenspace Similar Matrices Diagonalization Complex Eigenvalues and Eigenvectors 7: Orthogonality Norm of a Vector, Unit Vector, Standard Unit Vector Distance in R^n Dot Product Orthogonality of Vectors Orthogonal Projection Linear Combination of Orthogonal Vectors Gram Schmidt Process Orthogonal Matrices Orthogonal Diagonalization
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