Analysis of SOPs

The inverse word of "points" is "and", which is a word we are familiar. For example: 2 + 3 = 5, from left to right, we call the sum. In this case, if the score is a inverse word, the equation above is reversed: 5 = 2 + 3.

An object is expressed as the sum of two or more objects. This process is called analysis.

Generally, the analysis object should be consistent with the analyzed object. If it is a number, it is a number. If it is a function, it is a vector. If it is a matrix, it is a matrix.

Summation is the most basic operation in mathematics. subtraction, multiplication, and division are derived from summation. The more advanced power, finger, right, triangle, Calculus, etc. are also built on a layer-by-layer basis, and the most fundamental is the sum. The sum is the simplest, the easiest to calculate, and the simplest in nature. Therefore, it becomes the basic starting point of analysis.

The beauty of analysis is that through analysis, you can divide complex objects into simple objects. For example, 2 and 3 are simpler than 5. Study the properties of 2 separately, study the properties of 3 separately, and then grasp the properties of 5 through Simple summation. Divide complicated things into the sum of several simple objects, and break through simple objects. In addition, complicated things will be mastered.

Analysis is a fundamental part of Western thinking. Westerners believe that things always have a cause and effect. When we see the results, we need to analyze the causes. The reason for the analysis is to find out a bunch of factors, which indicate that these factors combined lead to the results. Westerners believe that things can always be analyzed. When we see the whole, we need to analyze the parts that are merged into the whole one by one. This is the case for a large part of modern science.

In college mathematics, there is a lot of content in analysis. The analysis in mathematics also expands the meaning, that is, to reasonably express a mathematical object as the sum of some simpler objects and the product of the real number coefficients. However, calculus and linear algebra have different focuses. Calculus studies the sum of infinity. The sum of infinity and infinity is essentially different. However, the sum of infinite items cannot be calculated, at least not actually. So we need to find a way to replace the approximation with the sum of the finite items. This is an approximation. The condition for establishing an approximation is convergence. That is to say, an approximation can be considered only when a part is cut out from the beginning of an infinitely converged entry to sum. The Chinese mathematician Xiang Wuyi said that calculus approaches this axe, but it is not always negative.

Calculus mainly studies functions. the dependent variable Y of continuous functions will change due to the change of the independent variable X. This change also needs to be analyzed. When X is changed from x0 to X1, how does y change from y0 to Y1? According to the above statement, "y change (y1-y0)" is a mathematical object, with a series of relatively simple "change" to represent. Mathematicians have found a sequence of converged "changing" objects. The first place is a linear variation, and its coefficient is the derivative, which is itself a differential dy. Mathematicians also found that when X's variation is infinite hours, as long as the first item, that is, the differential dy, is cut out from this infinite, converged "change" Object Sequence, in any case, you can accurately describe the changes of Y. I have seen such a statement in a book. Taylor's formula is the top of mathematical analysis. I don't know if it makes sense. I think so. With the Taylor formula, we can accurately calculate the value of a function at a certain point. After all, it's just the sum of real numbers.

However, in order to represent Taylor's formula, we used a complex concatenation algebraic formula. The Algebraic form cannot simply add up to an object as a real number. It can only represent the form of integration. This is how we realize that there are some special differences between objects in this concatenation, so that they cannot be simply added together. Therefore, it is necessary to discuss the nature of the object formed by adding something of different nature. This is the vector.
 
Calculus studies how to break down an object into the sum of Infinitely homogeneous objects, and linear algebra studies the nature of the new object "The sum of finite heterogeneous objects. On the one hand, as mentioned above, calculus still needs to be converted to infinity and precision to approximation at the end. On the other hand, heterogeneous objects can be converted to homogeneous objects after some processing. For example, the power function of different times is a heterogeneous object, but once a specific value is substituted, it can be converted to a real number and become a homogeneous object. Therefore, the study of linear algebra is very important to calculus. Therefore, I think linear algebra and calculus should be discussed first in college.

Our Calculus Teaching focuses too much on the calculation of differentiation and points. In fact, what is more important in practice is what we call "series. That is to study how to express a table as a number-level series and a function as a function-level series.

Linear Algebra uses the sum of heterogeneous objects (vectors) as the basis of research. It studies what these newly defined objects can add up and represent. The conclusion is that a finite number of vectors can be connected together, that is, all vectors of the same dimension can be expressed as the sum of these vectors. Such a group of vectors with full presentation capabilities is linear-independent vectors that constitute a vector space, and they constitute a group of bases in this vector space.

Back to the concept of analysis, a vector can always be expressed as the sum of several same-order vectors, which is the analysis of vectors. However, not all of these analyses share the same value. In a certain operation, a special analysis can provide excellent performance, thus greatly simplifying the operation. For example, in most cases, when a vector is expressed as the sum of a group of orthogonal base vectors, the computation can be particularly convenient. In the face of a problem, find the most advantageous form of analysis, and reasonably represent the object to be studied as the sum of the product of the base object and the real number coefficient with special properties, this is an important step in analysis and the key to success. In this representation, coefficients are called coordinates.

The classic method is based on finding a group of excellent features. For example:

Fourier analysis is based on orthogonal functions, so it has excellent properties. since 1904, it has replaced the power function system and became the mainstream analysis.

In curve and surface fitting, orthogonal polynomial sets constitute the best base function. The Laplace interpolation polynomial has a special property, that is, 1 on the current node and 0 on other nodes.

The shape function in the finite element is similar to the Laplace interpolation polynomial.

The primary vibration type Superposition Method in structural dynamics is based on the orthogonal primary vibration type, and the displacement of Multi-particle system is analyzed.

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Many concepts in linear algebra are also related to points. For example, orthogonal vectors and orthogonal functions are the same thing. Can this analogy be extended? Maybe linear transformation and integral transformation are also similar? To be considered.