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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 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,

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

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

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

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 its transpose matrix a^t.The A determinant i

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

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

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

On the Internet to see an article, looked after the feeling quite deep. He tells the nature of linear algebra, and intuitively describes linear spaces, vectors, and matrices. The linear algebra course, whether you start from the determinant or directly from the matrix, is ri

The nature of linear algebraWen/FengThe linear algebra course, whether you start from the determinant or directly from the matrix, is riddled with inexplicable beginnings.For example, in the national General Engineering Department of teaching the most widely used in Tongji linear a

Linear algebra (Linear Algebra) This discipline everyone is not unfamiliar, if someone still feel a little rusty, then "determinant", "matrix" These concepts you should always have the impression. Each major of the university will study this course in different shades, the liberal arts generally put in the "University

the surface of the applications. Here's a series (still in progress) on the mathematics behind linear programming. The primary technique for solving them, called the simplex algorithm, was essentially a beefed up Gaussian elimination. error Correcting codes: Another unseen but widespread use of linear algebra are in

system, but the problems we need to solve, such as area calculation, image rotation, stretching, and other applications are irrelevant to the coordinate system, creating a virtual coordinate system often does not help solve the problem.
The vector model overcomes the disadvantages of the analytical model. If the analytical model represents a "absolute" world view, then the vector model represents a "relative" world view, I recommend that you think of

inner product can also be applied naturally to any dimension of space.With the concept of distance, of course we have to do something interesting with it. or from the concept of geometry, we can use distance to calculate the perimeter and area of a plane graph and the volume of the spatial graph. For example, we can calculate the coordinates of a point in a line or a plane that is closest to the nearest point on the straight or flat plane. The two applicati

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