From 5e109dd831bdc995bf612756b795abae157bd245 Mon Sep 17 00:00:00 2001 From: bovem Date: Tue, 2 Jul 2024 19:49:54 +0530 Subject: [PATCH 1/4] Added article on tensors --- README.md | 2 + content/contents.md | 2 + content/posts/linear-algebra/_index.md | 5 + .../linear-algebra/tensors/3d-tensor.png | Bin 0 -> 133795 bytes content/posts/linear-algebra/tensors/index.md | 106 ++++++++++++++++++ .../posts/linear-algebra/tensors/matrices.png | Bin 0 -> 40599 bytes .../tensors-matrices-vectors-venn-diagram.png | Bin 0 -> 87711 bytes .../posts/linear-algebra/tensors/vectors.png | Bin 0 -> 36741 bytes 8 files changed, 115 insertions(+) create mode 100644 content/posts/linear-algebra/_index.md create mode 100644 content/posts/linear-algebra/tensors/3d-tensor.png create mode 100644 content/posts/linear-algebra/tensors/index.md create mode 100644 content/posts/linear-algebra/tensors/matrices.png create mode 100644 content/posts/linear-algebra/tensors/tensors-matrices-vectors-venn-diagram.png create mode 100644 content/posts/linear-algebra/tensors/vectors.png diff --git a/README.md b/README.md index 7164c0e..ebf529b 100644 --- a/README.md +++ b/README.md @@ -27,6 +27,8 @@ For a glimpse into my journey and the projects I've been involved in, feel free - File Handling in Go - Concurrency in Go - REST API Requests in Go +- Linear Algebra + - Tensors, Vectors, and Matrices - Data Structures and Algorithms - Time Complexity - Arrays, Strings, and HashMaps diff --git a/content/contents.md b/content/contents.md index 5443319..975340b 100644 --- a/content/contents.md +++ b/content/contents.md @@ -26,6 +26,8 @@ summary: "Index of all content" - File Handling in Go - Concurrency in Go - REST API Requests in Go +- Linear Algebra + - Tensors, Vectors, and Matrices - Data Structures and Algorithms - Time Complexity - Arrays, Strings, and HashMaps diff --git a/content/posts/linear-algebra/_index.md b/content/posts/linear-algebra/_index.md new file mode 100644 index 0000000..262243e --- /dev/null +++ b/content/posts/linear-algebra/_index.md @@ -0,0 +1,5 @@ +--- +title: Linear Algebra +summary: Contains posts related to `Linear Algebra` +description: Contains posts related to Linear Algebra +--- diff --git a/content/posts/linear-algebra/tensors/3d-tensor.png b/content/posts/linear-algebra/tensors/3d-tensor.png new file mode 100644 index 0000000000000000000000000000000000000000..edac7918069a046fa9ee0f11b32ed5dd583d351b GIT binary patch literal 133795 zcmeFac{G)O_dkrtln_xFqfm*8$b3?0kRfB{B4a4?%qa~h%9vSHDs!gHR1(TOgbbNE zB6Q50-@cr4s{8Z1@9+2i<5}x@*0Z`-Yu%sgbe-#ZzxVLkd%yO6p90iWl;~*o(2$Uj z(49SV`XUJls^&NJz>4@gJxfGF~MG1S=x=l;W@$T1DXP5<+b zGu+&|s(vFE|APp*bzQgqC)JRVR(X&KM(y|yi>fLX`47?~-$6yy!DLbt{vQ@a)e-+6 zq{sbNr37jJRjJ?q(_VUv>2X2==dNA5xVRdxDbO+!?b=CzTX%+`?&_az`%%-x%S$>i z`O%|C$~yVGh~|o-1l-vE&_MnVH$#7-Ajd?9=|3b`d(%rllQNA1oxh$B?#_K4d?~WK zgNNWT`*r|`3_2VoJ{S4wG5E}a+;%s?Q&A4^6$cU5Kfxkt5nkWNrjI7R>KVN1{E5Op 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b/content/posts/linear-algebra/tensors/index.md @@ -0,0 +1,106 @@ +--- +author: "Avnish" +title: "Tensors, Vectors, and Matrices" +date: "2024-07-03" +description: "A tensor is an array of data expanding in multiple independent dimensions." +tags: ["math", "linear-algebra", "tensors", "vectors", "matrices"] +categories: ["Math", "Linear Algebra"] +series: ["Linear Algebra"] +aliases: ["tensors-vectors-matrices"] +ShowToc: true +TocOpen: false +comments: false +math: true +--- + +In linear algebra, a **tensor** is an array of data expanding in multiple (or zero) independent dimensions. It is used to represent quantities/equations/functions with multiple components, for example, the equation $3x+2y=0$ could be represented with the tensor $[3\ 2\ 0]$ where each value in the tensor represents the different components of the equation. + +The number of independent dimensions of a tensor is called its **rank**. + +Vectors and matrices could be generalized with the term tensor. The following Venn diagram visualizes the connection between them. + +

Venn Diagram of Tensors, Matrices and Vectors

+ +# Scalar (Rank 0 Tensor) +A rank 0 tensor does not expand in any dimension, it is used to represent quantities that could be expressed by just one component i.e. its magnitude or scale. For example, the distance of $8\ cm$ between two points could be represented by the tensor $[ 8 ]$. + +Since the value of a rank 0 tensor signifies only its *scale* or *magnitude* it could be called a **scalar** value. A scalar value could represent the mass of an object, the temperature of a room, the speed of a car, etc. + +# Vector (Rank 1 Tensor) +A rank 1 tensor (or a **vector**) expands in one dimension i.e. it represents values with more than one component. + +

Vectors (Rank 1) Tensor

+ +Vectors are used to represent the magnitude and direction of different components of quantities such as displacement of an object, velocity of a car, electric field generated by a particle, etc. For example, displacement of $5\ m$ in the east direction and $10\ m$ in the north direction for an object could be represented with the vector $[5\ 10]$ where each component represents displacement in the east and north direction respectively. + +In equations, vectors are denoted as bold letters ($\textbf{E}$) or letters with an arrow on top ($\vec{E}$). + +## Magnitude of a Vector +To obtain the magnitude/scalar value of a vector we have to square all of its components and take the square root of their sum. For example, the magnitude of the displacement vector in the example above i.e. the distance between the start and the end will be $$|\ [ 0\ 5 \ 0 \ 10]\ | = \sqrt{0^2 + 5^2 + 0^2 + 10^2} = 25+100 = 125\ m$$ +The magnitude of a vector is denoted by enclosing it within $|\ \ |$, for example, $|\vec{E}|$. + +# Matrix (Rank 2 Tensor) +A rank 2 tensor (or a **matrix**) expands in two independent dimensions. +

Matrices (Rank 2) Tensor

+ +A *system of linear equations* +$$x - 2y = 6$$ +$$ x - y = 4 $$ +$$ x + y = 0 $$ +could be represented using a matrix as +$$ \begin{bmatrix} 1 & -2 & 6 \\ 1 & -1 & 4\\ 1 & 1 & 0\end{bmatrix}$$ +This matrix has three rows and three columns. Thus, its **order** will be $3 \times 3$. +A matrix with one row or one column could be called a matrix or vector interchangeably. + +# Tensors (Rank >=3) +The following diagram visualizes a rank 3 tensor in a 3-dimensional plane. +

Rank 3 Tensor

+ +Tensors with the rank greater than 3 are difficult to visualize but they could be represented as nested arrays in any programming language. +```python +# The following array has four levels of nesting +rank_4_tensor = [ + [ + [ + [ + 6, 7, 8 + ], + [ + 8, 58, 26 + ] + ], + [ + [ + 5, 28, 19 + ], + [ + 10, 11, 12 + ] + ] + ], + [ + [ + [ + 13, 14, 15 + ], + [ + 16, 17, 18 + ] + ], + [ + [ + 19, 20, 21 + ], + [ + 22, 23, 24 + ] + ] + ] +] +``` + +# Resources +
What's a Tensor? +Tensor Rank +Part 1 : Linear equation of two variables and Matrices +Part 4B : Tensors, Scalars, Vectors, and Matrices diff --git a/content/posts/linear-algebra/tensors/matrices.png b/content/posts/linear-algebra/tensors/matrices.png new file mode 100644 index 0000000000000000000000000000000000000000..7bcaa0f303f4be1c813d67debf7ff895445fc304 GIT binary patch literal 40599 zcmeFZcR1C5_&<(}W91kflsIPg4v{@Wkv$WkBiTZ>V$trs$tH?~Y z-+g+2KfiTdzrTL}e!IG^_v?K+p0DTgzVGKf9{1xGrF%o2lI%Pg9v&X$H4PO#JiIfe zcz6VX2m<)zOS9!I_z%8^p1KlV#TTX(JUlesH5EkzANd%{+ z^T?MM4A^Wc(}m^p-xN1uza6T{KAK{xSBnxJm#S^cXXwS%1L?Oj5rfbhu z?7*O`{|x(-=Z+pq8;!ivbj>1Bs^#y*qC{%q;Spvg^6wJK(!uoh#+AyxsN>!GrL4nK zDF#F^-v9hn&?P-;&6QwUBgRMK{mlD-jtJ|R+ z+`QZ*ID#EiRII}6At-lFsm`yP5>yOhGqx(+2T=0Z)raqP?)L54N}VN;K%$9bP7&X& z_Ghw0qDWg+_s>X2FIMVLXYcNMwK#bmY*nzwsufg~@bdBX{9>RYs|2A$Ad*bU$uq4W% z{;xL32$s#g%eKVc={n#>qW+7PN z6C&w4`HRyk4!g_N#a^G{%)M_WoUBP* zdtR>Vi5Ghz34SGQf0Lqi-!^SA-eBUH)ReO0jZ~|LgVWQ39LTDT1!U+)s<-uu_ 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z0;c6vhgd2u_1NAOv@S832KZO204Lz>>}+8@zdZKOliib~S_wi0C|G%{Vy!0aARlU@ls|l^RUqI`FvN*E zJ{4KceuLG+x3zfHZrDAFHCZSYN&tlXwU)JZA_=B&Z0In*3w;x(sk$x&MIjI-L1A^O zjh6<@2}o(qcrls^iGt{zXZpVCvzlW+6y3q{K7H0BGPTc8HGo;dKP39uZ1p8nO-*q8 zbw8)cQZGcEtHF%>7uemF=fTq8v=$f?S-cpv0`27$I{a8avmaF8Ju~xO)_gs-2(ShyS#=DHToJsKJ(qDic%nAjBIB<0u(Pu>GVFHf z<<4b|YRW?oN&ibrt^ZcG#HyRc?V~XQy%BmhR;`n?}a^>9F!C(-uLPV`a8r$SJeV{=eyZHm-YWU;|#!>hHfeR_K zeq@oleW#^mTj|j+%=3L(<3;yiYSbt9YO;G}w9p;%3&65=0TcjH9|_*i){~X>79*Lce0TQnU$A|v(l3G^5!Fdfw}=26D#&BQ}dGW7qp;v4LL z-V<2_Hehj;w8Fx1km&4)eh{}uYzv2eQgzT`P;~qPR*Oo6&OkkSvKDdihNXN-=5#7c zYS=4Cb~nX&_gnA{wh}y;14Hov?mqd(C&VXbV(P3O#ina%jQBL=E9^bRmN`i??52gOJ}jka+TgsD9-T@cxp zlIL-pt?=02U=QLH^$n&rw?A>i Date: Tue, 2 Jul 2024 20:32:01 +0530 Subject: [PATCH 3/4] Updated the name of article --- README.md | 2 +- content/contents.md | 2 +- content/posts/linear-algebra/tensors/index.md | 4 ++-- 3 files changed, 4 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index ebf529b..57eeaa0 100644 --- a/README.md +++ b/README.md @@ -28,7 +28,7 @@ For a glimpse into my journey and the projects I've been involved in, feel free - Concurrency in Go - REST API Requests in Go - Linear Algebra - - Tensors, Vectors, and Matrices + - Tensors - Data Structures and Algorithms - Time Complexity - Arrays, Strings, and HashMaps diff --git a/content/contents.md b/content/contents.md index 975340b..ee17ef2 100644 --- a/content/contents.md +++ b/content/contents.md @@ -27,7 +27,7 @@ summary: "Index of all content" - Concurrency in Go - REST API Requests in Go - Linear Algebra - - Tensors, Vectors, and Matrices + - Tensors - Data Structures and Algorithms - Time Complexity - Arrays, Strings, and HashMaps diff --git a/content/posts/linear-algebra/tensors/index.md b/content/posts/linear-algebra/tensors/index.md index d1f7dfe..01e2b19 100644 --- a/content/posts/linear-algebra/tensors/index.md +++ b/content/posts/linear-algebra/tensors/index.md @@ -1,12 +1,12 @@ --- author: "Avnish" -title: "Tensors, Vectors, and Matrices" +title: "Tensors" date: "2024-07-03" description: "A tensor is an array of data expanding in multiple independent dimensions." tags: ["math", "linear-algebra", "tensors", "vectors", "matrices"] categories: ["Math", "Linear Algebra"] series: ["Linear Algebra"] -aliases: ["tensors-vectors-matrices"] +aliases: ["tensors"] cover: image: "tensors-cover.png" linkFullImages: true From f73e6dc026ad5b867a302ad0fb12217c3adf1ed8 Mon Sep 17 00:00:00 2001 From: bovem Date: Tue, 2 Jul 2024 21:17:05 +0530 Subject: [PATCH 4/4] Fixed matrix rendering --- content/posts/linear-algebra/tensors/index.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/content/posts/linear-algebra/tensors/index.md b/content/posts/linear-algebra/tensors/index.md index 01e2b19..b1430fa 100644 --- a/content/posts/linear-algebra/tensors/index.md +++ b/content/posts/linear-algebra/tensors/index.md @@ -55,7 +55,9 @@ $$x - 2y = 6$$ $$ x - y = 4 $$ $$ x + y = 0 $$ could be represented using a matrix as -$$ \begin{bmatrix} 1 & -2 & 6 \\ 1 & -1 & 4\\ 1 & 1 & 0\end{bmatrix}$$ + +$$ \begin{bmatrix} 1 & -2 & 6 \\\ 1 & -1 & 4 \\\ 1 & 1 & 0 \end{bmatrix}$$ + This matrix has three rows and three columns. Thus, its **order** will be $3 \times 3$. A matrix with one row or one column could be called a matrix or vector interchangeably.