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matrix A and e just can reflect these 3 elementary transformations.The proof that the determinant is equal to the original determinant:This problem is very simple, but we should be able to realize the meaning of this theorem, it makes the row transformation and column transformation has the equivalence, that is, the application of the transformation of the row is applicable to the column.A brief proving process: defining determinant A and writing out

rate, that is, AB! = BA.@, define a m*n matrix a transpose (transpose) to n*m Matrix B, defined as Bji = AIJ where j=1, ..., N and I=1, ..., M. a transpose is at.@, defines a n*n matrix A, if satisfied at=a, is called symmetric (symmetric).@, the Law of matrix algebra:@, Unit matrix (Identity matrix): I = (Δij), where Δij = 1 (When I = j) or 0 (when i≠j). BI = B, IC = C.@, use column vectors to denote i, I = (e1, E2, ..., en), do not use IJ.@, define

The two theorems are very simple, obviously, and seem to be the basic arithmetic in proving matrix algebra. But it lays a theoretical foundation for understanding Matrix-vector Product ax with "linear transformation".Combining the properties of the matrix and vector product AX we discussed before, we can introduce a linear transformation.Because the properties of

we are faced with the question of how do we determine the linear correlation for a given m-vector r^n?There is a definition of what kind of algorithm, through the beginning of our definition of linear correlation, we can find that we only need to discuss the vector equation x1v1 + x2v2 + x3v3 +...+XMVM = 0 solution can, this goes back to our previous section introduced the use of the Jingzhen matrix to sol

Linear transformations:As we mentioned earlier, when we discuss the matrix equation ax = b and the vector equation X1a1+x2a2+x3a3+...+xnan = b, we have said that this will echo the rules of matrix multiplication. But here we first introduce a concept of transition-the linear transformation.To investigate the matrix equation ax = B,a is the n x m matrix, X is the r^n vector, by the rule we defined earlier, B

the determinant of the time will give a detailed proof.The other part is the general algorithm used when solving the inverse matrix of 3 order and above.First we give a lemma:Theorem 1: if n x n matrix A is reversible, then for any r^n vector B, the solution to the matrix equation ax = b is only present.Proof: existence, in this matrix equation is multiplied by the inverse matrix of a, then there is x = a^-1 B. Uniqueness, combined with the properties of the inverse matrix uniqueness mentioned

the linear Equation Group general solution, This allows us to directly transform the vector equation into an augmented matrix to solve the problem.Meaning of the span symbol: based on a linear combination of concepts, we remember that span (V1,v2,v3,..., vn) represents a collection of all linear combinations of v1~vn of these n vectors.Based on this concept of t

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

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,

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

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

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

Based on the previous chapters, we can easily draw the concept of eigenvectors and eigenvalues.First we know that the product of a and n dimensional vector v of n x n matrices will get an n-dimensional vector, then we now find that, after calculating u=av, the resulting vector u is collinear with V, that is, vector v is multiplied by matrix A to get the vector u "stretched" with respect to vector V, which satisfies the following equation:Av =λv=uSo here we call λ the eigenvalues of matrix A, and

Clairvoyant-Linear algebra-matrix theoryBook recommendations:Linear algebra: Domestic I think Li Shangzhi's linear algebra and blue to the high-generation concise tutorial is very good, the concept of explanation is very easy to understand, the study of computational skills

tell him loudly that the essence of the matrix is the description of the motion. But how interesting is it that the vectors themselves can also be viewed as n x 1 matrices? It is really fascinating that the object and movement in a space can be expressed in a similar way . Can you say it's a coincidence? If It's a coincidence, that's a lucky coincidence! It can be said that most of the wonderful properties of linear

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Can be deduced from 5 a^-1 = 1/det (a) * adj a. Adj a = [( -1) ^i+j* det (Aji)]
The relationship between determinant and volume: the area or volume of parallel geometry equals |det (A) |. and det (Ap) = det (A) *det (p)
Section Fourth: Vector Spaces
An indexed set {v1, v2, ... vp} of or more vectors, with VI! = 0, is linearly dependent, if and only if some VJ (w ITH J > 1) is a linear combination of the preceding vectors.

, it would be a lucky coincidence! It can be said that most of the wonderful properties of linear algebra are directly related to this coincidence.Then understand the matrix, which says "matrix is the description of motion", so far, it seems that everyone has no opinion. But I believe in the morning and evening there will be a mathematical department of the Netizen to make the decision turn. Because the con

§ 1 Elimination Method
I. Elementary Transformation of Linear Equations
Now we will discuss general linear equations. The so-called general linear equations refer to the form
(1)
Equations, which represent an unknown number, are the number of equations, called the coefficients of l

Review... Copy linear algebra and Its Application
Linear Equations
1. Similar
X_1-2x_2 =-1
-X_1 + 3x_2 = 3
There are three situations
1. No solution 2. There is a unique solution 3. There is an infinite Solution
Consider two parallel lines, the intersection line, and the exact coincidence line. Solving Equations
Primary Line Transformation (Multipl

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