3.3 What are eigenvalues and eigenvectors?

Let's just say now: In which direction is the feature vector block stretched, and how many times in each direction. This also makes it easy to understand why the value of the determinant is the product of the eigenvalues.

The eigenvectors also represent some good properties, that is, the lines do not shift in direction (can be reversed) after linear transformation, but only change in length.

4 Basic concepts of linear algebra

Why study mathematics? Why learn line generation? - Zhihu Big Question: What is the use of learning mathematics? What is the use of learning linear algebra? Trivia question: major, mathematics, computer programming, which one should I learn? Computational chemistry is the same as other calculations. The most important thing is not to master the programming language, but to construct a chemical problem into a computable mathematical problem, right? This is the heaviest... https://zhuanlan.zhihu.com/p/586540676

General mathematics, the study of relationships between numbers, with some numbers replaced by variables

In linear algebra, study the relationship between arrays--vectors y=ax where y, a, and x are all arrays, not just a is an array

RGB color is a way of expressing colors with arrays instead of numbers

A and B are two N arrays of the same order and rank. They seem to have the same structure, but when they are multiplied, they are on the right side, and the status is very different.

On the left, you are the base, the foundation of space, the coordinate system, the God Needle for Dinghai where you can go and where you can go, and the Tathagata's bergamot; Somersault, or eleven turns (you can't get out of the Tathagata Buddha's palm, right)? No matter how many changes are made on the right side, the result of the toss is in the space framed on the left side.

Matrix multiplication, on the left and on the right, has different meanings - Zhihu Supplement 2 (20220102) The original text only explains from the algebraic point of view that the matrix multiplication caused by the different meanings of the matrix on the left and right is not interchangeable. This supplement starts from the left and right matrix arrays ( Column vector) to explain the different entity properties. The biggest difference between line algebra and function is that function studies the relationship between numbers, line… https://zhuanlan.zhihu.com/p/166080173 What is the essence of matrix multiplication? - Knowing that the multiplication of matrices is essentially a kind of exercise. I offer here a model that I think is instructive to illustrate why matrix multiplication is a movement. 1 line… https://www.zhihu.com/question/21351965/answer/204058188

I have always wondered why the multiplication of matrices is defined in this way, and why such a weird multiplication rule can play such a huge role in practice? - Knowing that everyone has talked so much, I will give you an example of "matrix" used in the real world: ====================== ===========… https://www.zhihu.com/question/30898332/answer/2687307391

Let’s talk about bases and dimensions first. There must be a basis in a linear space, and any element in the linear space can be linearly expressed by the basis, and the way of expression is unique, and the unique combination of expression is the coordinate of this element under this basis. What is the necessary and sufficient condition for the linear expression and the only express way? Here, the concept of linear independence and maximum linear independent group is introduced , and the number of elements of the maximum linear independent group can lead to the concept of rank. The concept of dimension can be derived from the rank. The above concepts are all derived to describe the basis and dimension of linear space, and they do not appear out of thin air.

Next, let’s talk about isomorphism. There are tens of thousands of linear spaces, how should we study them? Isomorphism is such a powerful concept. Any linear space with the same dimension is isomorphic. The dimension of space is simple and profound. Simple natural numbers can describe the most essential properties of space. With the help of isomorphism, to study any n-dimensional linear space, one only needs to study Rⁿ.

As an n-dimensional linear space as a whole, we naturally think whether we can study its local properties first? So the concept of subspace and the direct sum decomposition of the whole space are naturally derived. Direct sum decomposition requires decomposing the whole space into the sum of two disjoint subspaces, and by studying the properties of each simple subspace, the properties of the whole space can be obtained.

4.2 Linear Mapping

nuclear space

1) The kernel space of the linear map. This is an important concept of linear mapping. What is the kernel space of linear mapping? Simply put, it is the set of preimages mapped to zero, denoted as KER. Using the analogy of a proportional function, it is obvious that when k is not equal to 0, its kernel is the null space, and when k is zero, its kernel space is the entire R.

Sometimes it is necessary to determine whether a linear map is injective. According to the definition, it is not so easy to prove. At this time, we can judge from its kernel. As long as its kernel is zero, then this linear map must be injective.

2) The image of the linear map. When the independent variable is taken over the entire domain, its image value range becomes a linear subspace , called the image space, denoted as IM.

3) Matrix representation of linear maps. How should an abstract linear map be 'analytically' expressed? This expression is written out as a matrix, and this matrix depends on the choice of basis. That is to say, under different bases, linear maps have different matrices. There are infinite bases and corresponding matrices. This brings trouble to the study of linear maps with matrices.

Luckily we have the similarity matrix . The matrices of the same linear mapping in different bases are similar relations, and the similar invariants include rank, determinant, trace, eigenvalue, characteristic polynomial , etc. Therefore, the rank, determinant, trace, eigenvalue, and characteristic polynomial properties of linear maps can be studied through similarity matrices.

There are infinitely many matrices of linear maps, so which ones are worthy of attention? The first is the matrix under the orthonormal basis , which is also the most common.

However, the matrix of a linear map may be particularly complicated under the orthonormal basis, so it is necessary to select a set of special basis so that its matrix has the simplest matrix representation under this basis. A linear map is said to be diagonalizable if there exists a basis such that the matrix of the linear map is diagonal.

However, can all linear maps be diagonalized? Unfortunately, not. Then we have to ask, if a linear map cannot be diagonalized, what is its simplest matrix ? The answer to this question is the Jordan standard form . It can be proved that there is a unique Jordan canonical form for any linear mapping over the field of complex numbers .