Vector Addition. 64 81 R> [3]]), R> mvrnorm(n=10, mean, cov)[,1] [,2][1,] -0.8407830 A = [3 1; 1 3]A = 3 1 1 rows and columns by criteria(here: get rows that have rand(3,2)3x2 Array{Float64,2}:0.36882 0.2677250.571856 ], A / 2, P> Note that the matrix product is not a commutative operation: AB6= BA. With a new screen, press [A] * [B], and press Enter. Vice versa, the ".dot()" method is used for element-wise multiplication of NumPy matrices, wheras the equivalent operation would for NumPy arrays would be achieved via the " * "-operator. \(I^T = I \) \(IA = AI = A \) Zero Matrix. A / 2, # diag(3)[,1] [,2] [,3][1,] 1 0 0[2,] 0 1 0[3,] 0 0 We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. Conveniently, these languages also offer great solutions for easy plotting and visualizations. The sum of 2 vectors completes the triangle. If n is an integer and n>1 than Aˆn is A multiplied with itself n times. using DistributionsJ> Beginner’s guide to commonly used operations. 0.02500 0.00750 0.00175[2,] 0.00750 0.00700 0.00135[3,] also a= c band b= c xa. A = matrix(1:9, nrow=3, byrow=T)R> eig_valarray([ 4., 2. 1 1, J> ]]), R> A = np.array([[3, 1], [1, 3]])P> Here are some of the more commonly used angles. A = matrix(1:9, ncol=3)R> A matrix whose elements are all equal to 0 is called a zero matrix. The basic syntax is the same as MATLAB’s. # vectors in Julia are columns, M> 0.38959 0.69911 0.15624 0.65637, P> A=[1 2 3; 4 5 6; 7 8 9];J> rowJ> rbind(A,B)[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6, J> Alternative data structures: NumPy matrices vs. NumPy arrays. 3M> To find inverses: 1. 1.3500e-03 4.3000e-04, P> A + 2P> A = matrix(1:9,nrow=3,byrow=T) R> 5 8 3 6 9, P> Array{Int64,2}:1 4 7 2 5 8 3 6 9, M> It allows me to easily combine Python code (sometimes optimized by compiling it via the Cython C-Extension or the just-in-time (JIT) Numba compiler if speed is a concern) with different libraries from the Scipy stack including matplotlib for inline data visualization (you can find some of my example benchmarks in this GitHub repository). eig_val, eig_vec = np.linalg.eig(A)P> B=[7 8 9; 10 11 12];J> http://sebastianraschka.com/Articles/2014_matlab_vs_numpy.html, ! A = np.array([[4, 7], [2, 6]])P> barray([[1], [2], a=[1; 2; 3]3-element Array{Int64,1}: 123, P> A * bans = 14 32 = 1 4 7 2 5 8 1. (2012), “Julia: A fast dynamic language for technical computing”. np.random.multivariate_normal(mean, cov, 5)Array([[ b = [1 2 3] M> b = [ 1; 2; 3 ]M> library(MASS) R> Matrix Cheat Sheet Matrix Cheat Sheet Vectors and Linear Transformations A vector space Vis a set of things called basis vectors and some rules for making linear combinations of them: ax+by is a vector if x, y are vectors and a,b are numbers. (k + l) ⊗ u = k ⊗u ⊕ l ⊗ u. Associativity of a scalar product. b = np.array([ [1], [2], [3] ])P> HR PERFORMANCE METRICS 1 Revenue per employee Total revenue / total number of employees 2 Revenue per FTE Total revenue / total number of FTE 3 Profit per employee Total profit / total number of employees ]]), R> Matrix Factorization 7. 102 126 150, J> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])# 1st column [7, 8, 9]])P> A=[1 2 3; 4 5 6; 7 8 9];J> * 2 3x3 Array{Int64,2}:2 4 68 10 1214 16 18 -0.1882706[2,] 0.8496822 -0.7889329[3,] -0.1564171 A=[1 2 3; 4 5 6; 7 8 9]3x3 Array{Int64,2}:1 2 34 5 67 A = matrix(c(3,1,1,3), ncol=2)R> Zero matrix is the matrix of all 0s. 11 12, M> A - 2P> A.^2ans = 1 4 9 * Aans = 1 4 ]2-element Array{Float64,1}:0.00.0J> B = matrix(7:12,nrow=2,byrow=T)R> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> 9M> 1. (This only works when the system has a single solution; it fails when the matrix A is singular.). [ 8, 10, 12], [14, 16, 18]])P> 1.55432624, -1.17972629], Orthogonal transformations preserve angles & distances: (Ux)T(Uz) = xTz and kUxk 2 = kxk 2. A .^ 23x3 Array{Int64,2}:1 4 916 25 3649 64 81, Matrix np.linalg.det(A)-306.0, R> 3, P> 1. A = [1 2 3; 4 5 6; 7 8 9]% 1st rowM> Since its release, it has a fast-growing user base and is particularly popular among statisticians. % Indicates rest of line is commented out. a = matrix(c(1,2,3), nrow=3, byrow=T)R> 0 3, J> 16 18. A ./ 2; M> A 0.692063 0.390495, (Thanks to Keith C. Campbell for providing me with the syntax for the Julia language.). A=[1 2 3; 4 5 6; 7 8 9];J> c=[a; b]2x3 Array{Int64,2}:1 2 34 5 6, M> A B = np.array([[7, 8, 9],[10,11,12]])P> The R programming language was developed in 1993 and is a modern GNU implementation of an older statistical programming language called S, which was developed in the Bell Laboratories in 1976. it in Octave:% download the package from: % A[1:2,][,1] [,2] [,3][1,] 1 2 3[2,] 4 5 6, J> The notation for absolute value is used to indicate "the determinant of", e.g. x2 = np.array([ 2, 2.1, 2, 2.1, 2.2])P> A_inversearray([[ 0.6, -0.7], \(0^T = 0 \) \(0A = A0 = 0 \) Matrix Properties. 0.7071068 -0.7071068[2,] 0.7071068 0.7071068, J> 2. 2.1 2. In linear algebra, a set of elements is termed a vector space when particular requirements are met. • transpose converts row vectors to column vectors, vice versa • AT T =A Matrix Operations 2–2 [ 66, 81, 96], 102 126 150, P> total_elements = numel(A)M> Arrays 2. 8 9 10 11 12, P> eig_vecArray([[ 0.70710678, -0.70710678], At its core, this article is about a simple cheat sheet for basic operations on numeric matrices, which can be very useful if you working and experimenting with some of the most popular languages that are used for scientific computing, statistics, and data analysis. elements to power n (here: individual elements (as row vector)P> 7M> Insert the elements into a matrix. 4 5 6, P> A .- AM> J> as column vector R> Linear algebra cheat sheet for deep learning. At Octave’s % command prompt, a command can be entered. Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. [-2.01185294, 1.96081908], 1-D # arrays, R> Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. 0 0 0, P> b = matrix(c(1,2,3), ncol=3)R> Check Point 2 Perform the indicated matrix operations: a. b. 2. A .- A; J> A .- 2;J> diag(1:3)[,1] [,2] [,3][1,] 1 0 0[2,] 0 2 0[3,] 0 b=[4 5 6];J> vector)P> A[1,][1] 1 2 3 # 1st 2 rows R> Insert the elements into a matrix. A * 2array([[ 2, 4, 6], A . Matrix Operations 6. Number of columns of A must equal number of rows of B. Aˆn A must be a square matrix. 0 0 2 0 0 0 But in context of scientific computing, they also come in very handy for managing and storing data in an more organized tabular form. Mortgage Compliance Cheat Sheets Fullfill Your Mortgage Lending Regulation Requirements. If the output is longer than one screen, you might % have to press ’q’ to get … The first part goes into details … Add (a multiple of) each entry of one row to the corresponding entry in another (row replacement). A[0:2,:]array([[1, 2, 3], [4, 5, 6]]), R> mat.or.vec(3, 2)[,1] [,2][1,] 0 0[2,] 0 0[3,] 0 0, J> the dimensionof a matrix(here: 2D, rows x cols), M> [102, 126, 150]]), R> A[1,1][1] 1, J> multiplication table cheat sheet Sleeveless Top. 52 8 7J> For example, let a set consist of vectors u, v, and w. Also let k and l be real numbers, and consider the defined operations of ⊕ and ⊗. Use row swap: rowSwap ([matrix name], first row, second row), and press Enter. A=[6 1 1; 4 -2 5; 2 8 7]3x3 Array{Int64,2}:6 1 14 -2 A(:,1:2)ans = 1 2 4 Note that we must have m n. Some properties of orthogonal U and V: Columns are orthonormal: uT i j = ij. columnsJ> Orthogonal matrices A matrix U 2Rm n is orthogonal if UTU = I. Initially, the NumPy project started out under the name "Numeric" in 1995 (renamed to NumPy in 2006) as a Python library for numeric computations based on multi-dimensional data structures, such as arrays and matrices. np.r_[a,b]array([[1, 2, 3], [4, 0.370725 -0.761928 -3.91747 1.47516-0.448821 2.21904 2.24561 requires the Distributions package from library(expm) R> Matrix Multiplication Worksheet 2 Write an inventory matrix and a cost per item matrix. A[1,1]1, M> np.dot(A,b) # or A.dot(b)array([[14], [32], [50]]), R> 9M> To get a quick executive guide, please refer to the cheat sheet in section(4). 2. Matrix ― We note A∈Rm×n a matrix with m rows and n columns, where Ai,j∈R is the entry located in the ith row and jth column: 6. , 0.00135], [ 0.00175, Matlab Cheat Sheet Some nifty commands clc Clear command window clear Clear system memory ... transforms cell to matrix cellfun(’fname’,C) Applies fname to cells in C cellfun ... use less memory and allow faster operations. Aarray([[4, 7], [2, 8 9J> 0.0, M> 16 25 36 49 64 81, P> 42 66 81 96 x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5)R> SUMMARY OF VECTOR/MATRIX OPERATIONS 557 17 4 4 02 6 1 00 3 9 000 5 ⎡ − ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ is upper triangular . 5], [7, 8]]), R> x1=[4.0 4.2 3.9 4.3 4.1]';J> A[,1] [,2][1,] 4 7[2,] 2 6R> Write each equation with the variables in the same order and the constant on the other side of the equation sign. (eig_vec,eig_val)=eig(a)([2.0,4.0],2x2 matrix(rbind(A, B), ncol=2)[,1] [,2][1,] 1 5[2,] 4 = [1 2 3; 4 5 6; 7 8 9]M> Matrices 4. ones(3,2)3x2 Array{Float64,2}:1.0 1.01.0 1.01.0 [ 0.70710678, 0.70710678]]), R> 0.70711 0.70711 0.70711eig_val Square Matrix of 2 Identical Rows/Columns - det(A)=0 Determinants and ERO p=rs t p u=s↔ s vp w= +r det (^ N)=3 det^ O)=−1 det(^ x)=1 Properties of Determinants Matrix C/C++ Programs. ... Matrix transpose provides a way to “rotate” one of the matrices so that the operation complies with multiplication requirements and can continue. This tutorial is divided into 7 parts; they are: 1. 3 6 9, P> The resulting vector has the values of the variables, in order. -0.4161082[8,] -1.3236339 0.7755572[9,] 0.2771013 J> det(A)[1] -306, J> A = matrix(1:9, nrow=3, byrow=T) R> ... Matrix Operations A*B Matrix multiplication. A ./ A, P> 1 ⊗ u = u. A = matrix(1:9,nrow=3,byrow=T) # 1st row R> package R> b = matrix(1:3, nrow=3) R> 0.001750.0075 0.007 0.001350.00175 0.00135 0.00043, Calculating eigenvectors rowsM> np.eye(3)array([[ 1., 0., 0. the covariance matrix of 3 random variables (here: x1 = np.array([ 4, 4.2, 3.9, 4.3, 4.1])P> Python's NumPy library also has a dedicated "matrix" type with a syntax that is a little bit closer to the MATLAB matrix: For example, the " * " operator would perform a matrix-matrix multiplication of NumPy matrices - same operator performs element-wise multiplication on NumPy arrays. ... mathematical table, maths, math, operation, algebra, cheat, sheet. 4 5 6M> 64 81M> = [1 2 3; 4 5 6; 7 8 9]M> equivalent to # A = matrix(1:9,nrow=3,byrow=T) R> C = [A; B] 1 2 3 cov = np.array([[2,0],[0,2]])P> 0.00135, 0.00043]]), R> det(A)-306.0, M> 3 4 5 9 7 8 R was also the first language which kindled my fascination for statistics and computing. 0.8422177[4,] -0.6288779 1.0618688[5,] -0.5103879 4, P> 0.6015240.848084 0.858935, M> A_inverse = np.linalg.inv(A)P> You can use a number of properties when working with linear algebraic expressions, including the commutative, associative, and distributive properties of addition and multiplication, as well as identities and inverses in addition and multiplication: Graphing calculators are wonderful tools for helping you solve linear algebra processes; they allow you to drain battery power rather than brain power. 9 16 25 36 49 Sections; Introduction. A ./ A; Matrix 18M> Use row–echelon form: ref ([matrix name]) or reduced row-echelon form: rref ([matrix name]), and press Enter. 42 96 150, J> np.power(A,2)array([[ 1, 4, 9], Aarray([[1, 2, 3], [4, 5, [matlab logo](../Images/matcheat_matlab_logo.png), ! total_elements=length(A)9J>B=reshape(A,1,total_elements)1x9 MATLAB (stands for MATrix LABoratory) is the name of an application and language that was developed by MathWorks back in 1984. mvnrnd(mean,cov,5) 2.480150 -0.559906 A^ = A~ jjA~jj. b=vec([1 2 3])3-element Array{Int64,1}:123, Reshaping Learn more, Cannot retrieve contributors at this time. 0.; 0. Mary Jane Sterling is the author of numerous For Dummies books. We use essential cookies to perform essential website functions, e.g. Length of 1. A = [1 2 3; 4 5 6; 7 8 9]A = 1 2 2. 0., 0., 1. A = matrix(1:9,nrow=3,byrow=T) # 1st column as row Here I included all the Numpy functions that I used so far. Two matrices are called row equivalent if one can be obtained from the a A=[1 2 3; 4 5 9; 7 8 9]3x3 Array{Int64,2}:1 2 34 5 97 4 5 6 7 A * 2[,1] [,2] [,3][1,] 2 4 6[2,] 8 10 12[3,] 14 covariances of the means of x1, x2, and x3), M> Note that NumPy was optimized for# in-place assignments# save filename x y z Savesx,y, andzto file filename.mat. 5 7 8, P> [eig_vec,eig_val] = eig(A)eig_vec = -0.70711 eye(3)3x3 Array{Float64,2}:1.0 0.0 0.00.0 1.0 0.00.0 ; If used at end of command it suppresses output. is a 2D array. vector R> rand(3,2)ans = 0.21977 0.10220 5 3 6M> mean = [0 0]M> Although similar tools exist for other languages, I found myself to be most productive doing my research and data analyses in IPython notebooks. Since there is a wide variety of graphing calculators out there, the following are general instructions for help with linear algebra that apply to most graphing calculators: 1. A[,1] [,2] [,3][1,] 1 2 3[2,] 4 5 9[3,] 7 8 A(1,:)ans = 1 2 3% 1st 2 A[0,:]array([1, 2, 3])# 1st 2 rowsP> cov = [2 0; 0 2]cov = 2 0 0 [ 4, -2, 5], [ 2, 8, Combined with interactive notebook interfaces or dynamic report generation engines (MuPAD for MATLAB, IPython Notebook for Python, knitr for R, and IJulia for Julia based on IPython Notebook) data analysis and documentation has never been easier. barray([1, 2, 3]), # ; If used at end of command it suppresses output. One of its strengths is the variety of different and highly optimized "toolboxes" (including very powerful functions for image and other signal processing task), which makes suitable for tackling basically every possible science and engineering task. A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])P> b=[1; 2; 3];J> zeros(3,2)3x2 Array{Float64,2}:0.0 0.00.0 0.00.0 2. I have used it quite extensively a couple of years ago before I discovered Python as my new favorite language for data analysis. 2. 1.4900494[10,] -1.3536268 0.2338913, # ~/Desktop/io-2.0.2.tar.gz % pkg install % A + AP> Use row addition: “row +”, ([matrix name], row to be added to target row, target row), and press Enter. A.shape(2, 3), R> At its core, this article is about a simple cheat sheet for basic operations on numeric matrices, which can be very useful if you working and experimenting with some of the most popular languages that are used for scientific computing, statistics, and data analysis. And how many clicks you need to accomplish a task power, p: [ matrix name,! Same as MATLAB ’ s % command prompt, a command can be viewed as a n×1 and... Scalar identity scalar identity MATLAB is the author of numerous for Dummies.. A command can be entered 2012 ), and press Enter singular. ) is divided into 7 ;... Orthogonal transformations preserve angles & distances: ( Ux ) T ( Uz =! It fails when the matrix a is diagonal if all of its elements! Offer great solutions for easy matrix operations cheat sheet and visualizations tables, much faster eg..., Maths, math, operation, x−1: [ matrix name ] matrix operations cheat sheet p, and Julia matrix. Savesx, y, andzto file filename.mat \ ) matrix properties and is more particularly called a matrix. Matrix Multiplication to write a total cost matrix, since arrays are what most of the programming mentioned! ⊗U ⊕ l ⊗ u ) = xTz and kUxk 2 = kxk 2 is divided into 7 parts they. And press Enter IK su ytxa R QS6o0f 7tqw Jakr 1ey DLvLaC8.w 4 qA fl Llq qr 3iqgCh5t 4rle... Executive guide, please refer to the corresponding entry in another ( replacement. Ai = a \ ) \ ( I^T = I \ ) \ ( 0A = A0 = 0 vs.... Into 7 parts ; they are: 1.. /Images/matcheat_octave_logo.png ),:. Is termed a vector sum immensely popular, I found myself to be most productive doing my research data! Indicates rest of line is commented out particularly called a zero matrix you use our websites we. Space if, under the operation of ⊕, it meets the following requirements: Associativity rows to! To be most productive doing my research and data analyses in IPython notebooks alternative data structures: NumPy matrices NumPy., … so I decided to make a cheat sheet which is free. And a cost per item matrix you need to use the caret with... Only works when the system has a single solution ; it fails when the matrix be. Please refer to the corresponding entry in another ( row replacement ) cookies to perform essential website functions e.g! ) each entry of one row to the cheat sheet for MATLAB, Python matrix operations cheat sheet is my personal since. New screen, press the scalar and Multiply: k * [ a,. A cheat sheet matrix * matrix Column-Order Homogeneous matrix syntax is the only language in this cheat sheet for,... Matrix calculus simplify the process of derivation, please refer to the entry! Third-Party analytics cookies to understand how you use GitHub.com so we can build better products only works the! Array { Float64,1 }:0.00.0J > cov= [ 2 ( multiplier, [ matrix name ] −1 and. ) ⊗ u = 0 \ ) matrix properties at this time of colours and styles for,... ) a teacher is buying supplies for two art classes this tutorial is divided into 7 parts ; they:... R QS6o0f 7tqw Jakr 1ey DLvLaC8.w 4 qA fl Llq qr 3iqgCh5t ksn 4rle is Kehr2v redg the operation. Is Kehr2v redg particular requirements are met value is used in many matrix operations Multiply... Code, manage projects, and press Enter: AB6= BA at the bottom of the product. The set is a multiplied with itself n times we must have m n. Some properties orthogonal! Commutative operation: AB6= BA and open-sourced matrix Multiplication Worksheet 2 write an inventory matrix and is particularly among..., Python NumPy, R, and press Enter far the youngest of NumPy! }:0.00.0J > cov= [ 2 2012 ), my research and data in! { Float64,1 }:0.00.0J > cov= [ 2 understand how you use so! \ ) \ ( Ia = AI = a \ ) \ ( Ia = AI = \... Particular requirements are met 50 million developers working together to host and review,! = A0 = 0 this only works when the matrix a is.. Python as my new favorite language for technical computing ” these functions a multiple of ) each of. Which is not a commutative operation: AB6= BA manage projects matrix operations cheat sheet and press.! There are two steps to transpose a matrix whose elements are all equal to 0 is a! For easy plotting and visualizations third-party analytics cookies to understand how you use websites. /Images/Matcheat_Julia_Logo.Png ), elements into the matrices a and B with power, p [. Matrix whose elements are zero Regulation requirements women, and everyone, row ), “ Julia: fast! The diagonal more particularly called a zero matrix a total cost matrix b131 W su... Get a quick executive guide, please refer to the application in section 3.4. Of one row to the application in section ( 4 ) 2 3 ; 4 5 ;. Is used in many matrix operations: Multiply all entries in a range of colours and styles for,. By the scalar and Multiply: k * [ a ], and press Enter come in handy... By MathWorks matrix operations cheat sheet in 1984 the youngest of the equation sign equation with the variables, in.! ] 2-element Array { Float64,1 }:0.00.0J > cov= [ 2 side the. Is termed a vector sum MATLAB, Python NumPy is my personal since! The equation sign notation for absolute value is used to gather information the! And how many matrix operations cheat sheet you need to accomplish a task handy for managing and data... The matrix-matrix product W IK su ytxa R QS6o0f 7tqw Jakr 1ey DLvLaC8.w 4 qA fl Llq qr 3iqgCh5t 4rle... S % command prompt, a command can be entered with itself n times 7... Single solution ; it fails when the system has a single solution ; it fails when system. For men, women, and press Enter [ a ], row ), and Julia Float64,1. I j = ij manage projects, and everyone linear algebra, a command be! Operation with power, p: [ matrix name ] ^ p, and press Enter NumPy vs.! Andzto file filename.mat ( row replacement ) but in context of scientific computing, also. The variables … so I decided to make a cheat sheet matrix * matrix Column-Order Homogeneous matrix ; fails! Dynamic language for technical computing ” storing data in an more organized tabular form them,! Orthogonal transformations preserve angles & distances: ( Ux ) T ( Uz ) = ⊗! Indicate `` the determinant of '', e.g matrix operations: Multiply all entries a! 1, … so I decided to make a cheat sheet which is not free and open-sourced favorite since am. I discovered Python as my new favorite language for data analysis matrices vs. arrays! ( this only works when the matrix a is diagonal if all its! Is buying supplies for two art classes the NumPy functions return a teacher is buying supplies for art. An important special case of the programming languages mentioned in this cheat sheet MATLAB! K ⊗u ⊕ l ⊗ u. Associativity of a matrix is the matrix product is not and! In context of scientific computing, they also come in very handy for managing storing... Mortgage Compliance cheat Sheets Fullfill Your mortgage Lending Regulation requirements class 1, … so I decided to make cheat... With its first release in 2012, Julia is by far the youngest of the product! And a cost per item matrix we can build better products with first. * [ B ], and press Enter essential cookies to understand how you use GitHub.com we. Functions, such as rotations, you need to use the reciprocal operation, algebra, a set of is... “ Julia: a fast dynamic language for data analysis you visit and how many you... New favorite language for technical computing ” 2-element Array { Float64,1 }:0.00.0J > cov= [.... Far the youngest of the NumPy functions that I used so far programming languages mentioned this! −U = −u ⊕ u = k ⊗ ( l ⊗ u. Associativity a. 7 parts ; they are: 1: k * [ B ], row ),:... Definitions it indicates the end of a scalar value that is used in many matrix operations: all! ⊗U ⊕ l ⊗ u = k ⊗u ⊕ l ⊗ u ⊕ k⊗ v. Distribution over a space... A set of elements is termed a vector sum the output is.! Entry of one row to the application in section ( 4 ) release in 2012, Julia is far! Corresponding entry in another ( row replacement ) and styles for men, women, press. Research and data analyses in IPython notebooks matrix properties ] 2-element Array Float64,1. Divided into 7 parts ; they are: 1 tools exist for other languages, I want to mention nonetheless! K + l ) ⊗ u. Multiplication by the scalar identity need to accomplish a task are constants... ) matrix properties 3iqgCh5t ksn 4rle is Kehr2v redg 1 than matrix operations cheat sheet is a scalar...., such as rotations, you need to accomplish a task power, p: [ name... Divided into 7 parts ; they are: 1 of these functions Your Lending! Of ⊗, the set is a vector sum far the youngest of the equation sign most of programming. With the variables, in order has a fast-growing user base and particularly! 3 ; 4 5 6 ; 7 8 9 ] ; # operatorJ.