portrait of linear algebra

1. Determinant1.1 Second-order determinant1.2 third-order determinant1.3 Number of reverse order1.4 N-Step determinant2. The nature of the determinantProperty 1 The determinant is equal to its transpose determinant.Property 2 swaps the determinant of two rows (columns), determinant.Property 3 The determinant of a row (column) in which all elements are multiplied by the same multiplier K, equals the number k multiplied by this determinant.Property 4 Determinant If there are two rows (column) elem

This chapter begins with an introduction to another basic concept in linear algebra-the determinant.In fact, like the Matrix, the determinant is also a tool for simplifying the expression polynomial, about the historical origin of the determinant, as the following introduction.In introducing the inverse matrix, we have mentioned that the second-order matrix has a corresponding determinant based on matrix A

algebra review, I'll be the using one index vectors. Most vector subscripts in the course start from 1.When talking on machine learning applications, sometimes explicitly say if we need to switch to, when we need to use The zero index vectors as well. Discussion of machine learning applications will be converted to subscript starting from 0.Finally, by Convention,use upper case to refer to matrices. So we ' re going-letters like a, B, c.and usually w

In this chapter we discuss the relationship between the vectors defined in the R^n space, which is generally orthogonal, then the orthogonal projection, the best approximation theorem, and so on, these concepts will lay the foundation for the optimal approximate solution of the ax=b of linear equations with no solution.Orthogonality:To give the simplest example, in a plane, if the two-dimensional vector's point multiplication is 0, then we can determi

3 31 2 4 % % The basis of zero space: the Code of this zero space is worth looking at, reflecting the basic idea of column meaning in linear algebra. You may not know what the code is. A simple statement may contain many operations. I like to give an example when I don't quite understand it. Let's take a look at the code in one sentence. Note that The column in the zero space indicates the

transformation.Matrix elimination Element Method:determinant TypeCalculation (0 descending order method)Other properties of the determinant:The law of ClydeMatrixFollow the law1. Linear Properties2. Operational and polynomial of n-order matricesElementary matrix and its role in multiplicationFor the unit matrix, the matrix obtained by making an elementary transformation becomes the elementary matrix.Together there are three primary transformations:Th

Reprinted from: http://www.read.org.cn/html/2070-zhuan-zai-shi-tian-nei-zhang-wo-xian-xing-dai-shu-jing-ren-de-chao-su-xue-xi-shi-yan.html I just saw a very wonderful article in the translation, which was specially reproduced for sharing by more people. Source: calnewport.com Original article title: Mastering linear algebra in 10 days: astounding experiments in ultra-Learning Address: http://calnewport.

Prepare to write an article about Singular Value DecompositionArticleIt suddenly found that it needed a lot of linear algebra knowledge. Therefore, we will first introduce the basic concepts and operations of linear algebra to help readers understand Singular Value Decomposition. 1. Basic Concepts A matrix is a numbe

A very important concept in linear algebra is the vector space r^n, which will focus on a series of properties of vector space.A vector space is a non-empty set v consisting of some vector elements, which needs to satisfy the following axioms:The subspace H of the vector space V needs to meet the following three conditions:The two theorems are in the elaboration of how to form subspace, and its proof simply

transformed in order to find a suitable linear combination(linear combination) that makes AX = bThe corresponding diagramVector b is a sum of two col vectorsHere again, when Vector x is taken, we can get the entire XY plane, meaning that whatever vector B can find the corresponding solution(not when two col vectors are parallel)* The practice of column picture does not seem to be emphasized in schools, but

Matrix equation:We have previously introduced the linear combination of vectors, the form of X1a1+x2a2+xnan, that we can use to express them with [] formulas. (This expression is sought for convenience and unity of computation), and we give the following definition to give another form of the linear combination of vectors.It can be seen that the right side of the equation, the form of a vector combination,

I. Five Representation Methods of Matrix Multiplication 1. General Form 2. Multiply the matrix and column vector 3. Multiply a matrix and a row vector 4. Multiply Matrix Blocks Ii. Matrix Inversion For phalanx, left inverse = Right Inverse Multiply the original matrix by its inverse matrix to obtain the Unit Matrix Several methods to determine whether it is reversible:1. the determinant is 02. the columns of the matrix are linear combinations of

in the European coordinate system are obtained after the X vector (also a coordinate point in the M coordinate) is left multiplied by m in the Custom space. Space coordinates are converted. If the implementation of the European coordinate transformation to the M coordinate system, can be on both sides of the same time left multiplied by a m of the inverse matrix M-1, (M-1) * m * x = (M-1) * B is X = (M-1) * B. After B is used, X can be obtained, and then the coordinate of point X in the M coord

be extracted by an array named key;The Savetxt (), Loadtxt () function can read and write a text file that holds a one-dimensional and two-dimensional array, outputs the text separated by the spacer, specifies the spacer by the delimiter parameter, and the default output is in the form of '%.18e ', separated by a space by default.4. Memory-mapped arraysCreates a memory-mapped array from a file that reads the specified offset data,> without reading the entire file into memory by Memmap ():FileNa

assume that the line expression is as follows:Then calculate the error function:The error function e is obtained by the deviation of the coefficient, a, a, and the 0:The value of the coefficients, a, a, a, and a linear expression is obtained by the above formula:The curves obtained by the least squares are as follows:Linear algebraic approach to least squares: Similarly, the expression of a fitted line is set to:The purpose of fitting is to make the

Least squares problem:Before the combination of orthogonal, subspace W, orthogonal projection, orthogonal decomposition theorem and best approximation principle in vector space, the least squares problem can be solved satisfactorily.First of all, we have to explain the problem itself, that is, in the process of production, for the giant linear systems ax=b, may be no solution, but we are urgently need a solution, satisfies the solution is the most app

- voidInsert (Char*R) { About intLen=strlen (s), p=RT; $ for(intI=0; i) - if(Ch[p][id (S[i])) -p=Ch[p][id (S[i]); - Else AP=ch[p][id (S[i])]=++CNT; +tag[p]=true; the } - $ voidBuild () { thequeueint>Q; the for(intI=0;i4; i++) the if(Ch[rt][i]) thefail[ch[rt][i]]=Rt,q.push (Ch[rt][i]); - Else inch[rt][i]=RT; the the while(!Q.empty ()) { About intx=Q.front (); Q.pop (); th

[find (1)]!=1){ thememset (Vis,0,sizeof(VIS)); num=0; the for(intI=2; i1; i++){94 if(Vis[find (i)])Continue; theVis[find (i)]=++num; the } thenum=0;98 for(intI=2; i1; i++) Aboutnum=num*Ten+Vis[find (i)]; -a.mat[id[num]][id[mem[t]]]+=1;101 }102 }103 }104 return; the }106 107 intMain () {108 #ifndef Online_judge109Freopen ("count.in","R", stdin); theFreopen ("Count.out","W", stdout);111 #endif thesc

] ofRec; V:Array[0..2002000] ofBoolean; N,m,i,j,k,l,st,ed,ww,top,tar,ans,x:longint;functionmin (aa,bb:longint): Longint;begin ifAa Thenexit (AA); exit (BB);End;procedureAdd (st,ed,ww:longint);beginInc (top); A[TOP].S:=St; A[TOP].E:=Ed; A[TOP].W:=ww; A[top].next:=B[st]; B[ST]:=top;End;functionBfs:boolean;varHead,tail,x,u:longint; Y:rec;beginFillchar (v,sizeof (v), false); Tail:=1; head:=0; d[st]:=1; V[ST]:=true; q[1]:=St; whileHead Do beginInc (head); x:=Q[head]; U:=B[x]; whileU>0 Do begi

1.The calculate the slope:the covariance of X and Y divided by the variance of X　　From NumPy import CoVslope_density = CoV (wine_quality["quality"],wine_quality["density"]) [0,1]/wine_quality["Density"].var () #cov ( X, y) is the function from NumPy, which returns a 2*2 Metric,.var () is Pandas function.2.To get the INTERCEPT:B = Y-ax (x and Y is the mean value of each column)Intercept_density = wine_quality["Quality"].mean ()-wine_quality["Density"].mean () * (Calc_slope (wine_quality[) Density