Then understand matrices.
The last article said that "matrix is a description of motion", so far, it seems that everyone has no opinion. But I believe that sooner or later there will be netizens from the mathematics department who will make a decision. Because the concept of motion is associated with calculus in mathematics and physics. When we study calculus, someone will always tell you that elementary mathematics is the mathematics of constants and statics, and advanced mathematics is the mathematics of variables and the mathematics of movement. It is spread by word of mouth, and almost everyone knows this sentence. But there seem to be not many people who really know what this sentence means. In short, in our human experience, motion is a continuous process, from point A to point B, even the fastest light needs a time to pass through the path between AB point by point , which is Brings the concept of continuity. And the matter of continuity cannot be explained at all if the concept of limit is not defined. The ancient Greeks were very strong in mathematics, but they lacked the concept of limits, so they couldn't explain movement. They were made to die by Zeno's famous paradoxes (the four paradoxes that the arrow doesn't move, Achilles can't outrun the turtle, etc.). . Since this article is not about calculus, I won't say much about it. Interested readers can go and read "Revisiting Calculus" written by Professor Qi Minyou. It was only after reading the opening part of this book that I understood the truth of the saying "advanced mathematics is the mathematics of sports".
However, in my "Understanding Matrix" article, the concept of "motion" is not continuous motion in calculus, but an instantaneous change. For example, at this moment at point A, after a "movement", it " jumps " to point B at once, which does not need to pass through any point between point A and point B. Such "movements", or "jumps", are contrary to our everyday experience. However, people who know a little common sense of quantum physics will immediately point out that quantum (such as electron) jumps on different energy level orbits, which happens instantaneously, and has such a transition behavior. Therefore, it is not that there is no such movement phenomenon in nature, but we cannot observe it on a macro level. But in any case, the word "movement" is used here, which is prone to ambiguity. To be more precise, it should be "jump". So this sentence can be changed to:
"A matrix is a description of transitions in linear space".
But this is too physical, that is, too concrete, and not mathematical enough, that is, not abstract enough. So we finally use a real mathematical term - transformation , to describe this thing. In this way, everyone should understand that the so-called transformation is actually the transition from one point (element/object) to another point (element/object) in space . For example, topological transformations are transitions from one point to another in topological space. For another example, affine transformation is the transition from one point to another in affine space. By the way, this affine space and vector space are brothers. Friends who do computer graphics know that although only three-dimensional vectors are needed to describe a three-dimensional object, all computer graphics transformation matrices are 4 x 4. As for the reason, many books say "for convenience in use", which in my opinion is simply an attempt to get away with it. The real reason is because the graphics transformations applied in computer graphics are actually performed in affine space rather than vector space. Think about it, after moving in parallel with a vector in the vector space, it is still the same vector, and of course two parallel line segments of equal length in the real world cannot be regarded as the same thing, so the living space of computer graphics is actually a simulation. shoot space. The matrix representation of the affine transformation is simply 4 x 4. Again, interested readers can go to "Computer Graphics - Detailed Explanation of Geometric Tools and Algorithms".
Once we understand the concept of "transformation", the definition of a matrix becomes:
"A matrix is a description of transformations in linear space."
So far, we have finally got a definition that seems relatively mathematical. But a few more words to say. It is generally said in textbooks that a linear transformation T in a linear space V can be expressed as a matrix when a set of bases is selected. Therefore, we also need to make it clear what is linear transformation, what is basis, and what is a selected set of basis. The definition of linear transformation is very simple. There is a transformation T, such that for any two objects x and y that are not the same in the linear space V, and any real numbers a and b, there are:
T(ax + by) = aT( x) + bT(y),
then T is called a linear transformation.
Definitions are written in this way, but it is not intuitive to understand just by looking at the definitions. What kind of transformation is a linear transformation? As we just said, transformation is the transition from one point in space to another, and linear transformation is the movement from a certain point in a linear space V to another point in another linear space W. This sentence contains a layer of meaning, that is, a point can not only be transformed to another point in the same linear space, but also can be transformed to another point in another linear space. No matter how you change, as long as there are objects in linear space before and after the transformation, the transformation must be a linear transformation, and it must be described by a non-singular matrix. And a transformation you describe with a non-singular matrix must be a linear transformation. Some people may ask, why do we emphasize non-singular matrices here? The so-called non-singularity is only meaningful to the square matrix, so what about the non-square matrix? This will be more verbose. Finally, we need to use linear transformation as a mapping, and discuss its mapping properties, as well as concepts such as the kernel and image of linear transformation. I don't think this is the point. If you really have time, I will write more later. Below we only discuss the most common and useful transformation, which is the linear transformation within the same linear space. That is to say, the matrix mentioned below, unless otherwise specified, is a square matrix, and it is a non-singular square matrix. When learning a subject, the most important thing is to grasp the main content and quickly establish the overall concept of the subject. It is not necessary to consider all the details and special circumstances at the beginning, and make yourself a mess.
Next, what is a base? This issue will be discussed later, here as long as the basis is regarded as a coordinate system in a linear space. Note that it is a coordinate system, not a coordinate value, these two are a "unity of opposites and contradictions". Thus, "selecting a set of bases" means selecting a coordinate system in linear space. That's what it means.
Well, finally we perfect the definition of the matrix as follows:
"A matrix is a description of a linear transformation in a linear space. In a linear space, as long as we choose a set of bases, any linear transformation can be described by a definite matrix."
理解这句话的关键，在于把“线性变换”与“线性变换的一个描述”区别开。一个是那个对象，一个是对那个对象的表述。就好像我们熟悉的面向对象编程中，一个对象可以有多个引用，每个引用可以叫不同的名字，但都是指的同一个对象。如果还不形象，那就干脆来个很俗的类比。
比如有一头猪，你打算给它拍照片，只要你给照相机选定了一个镜头位置，那么就可以给这头猪拍一张照片。这个照片可以看成是这头猪的一个描述，但只是一个片面的的描述，因为换一个镜头位置给这头猪拍照，能得到一张不同的照片，也是这头猪的另一个片面的描述。所有这样照出来的照片都是这同一头猪的描述，但是又都不是这头猪本身。
同样的，对于一个线性变换，只要你选定一组基，那么就可以找到一个矩阵来描述这个线性变换。换一组基，就得到一个不同的矩阵。所有这些矩阵都是这同一个线性变换的描述，但又都不是线性变换本身。
但是这样的话，问题就来了如果你给我两张猪的照片，我怎么知道这两张照片上的是同一头猪呢？同样的，你给我两个矩阵，我怎么知道这两个矩阵是描述的同一个线性变换呢？如果是同一个线性变换的不同的矩阵描述，那就是本家兄弟了，见面不认识，岂不成了笑话。
好在，我们可以找到同一个线性变换的矩阵兄弟们的一个性质，那就是：
若矩阵A与B是同一个线性变换的两个不同的描述（之所以会不同，是因为选定了不同的基，也就是选定了不同的坐标系），则一定能找到一个非奇异矩阵P，使得A、B之间满足这样的关系：
A = P-1BP
线性代数稍微熟一点的读者一下就看出来，这就是相似矩阵的定义。没错，所谓相似矩阵，就是同一个线性变换的不同的描述矩阵。按照这个定义，同一头猪的不同角度的照片也可以成为相似照片。俗了一点，不过能让人明白。
而在上面式子里那个矩阵P，其实就是A矩阵所基于的基与B矩阵所基于的基这两组基之间的一个变换关系。关于这个结论，可以用一种非常直觉的方法来证明（而不是一般教科书上那种形式上的证明），如果有时间的话，我以后在blog里补充这个证明。
这个发现太重要了。原来一族相似矩阵都是同一个线性变换的描述啊！难怪这么重要！工科研究生课程中有矩阵论、矩阵分析等课程，其中讲了各种各样的相似变换，比如什么相似标准型，对角化之类的内容，都要求变换以后得到的那个矩阵与先前的那个矩阵式相似的，为什么这么要求？因为只有这样要求，才能保证变换前后的两个矩阵是描述同一个线性变换的。当然，同一个线性变换的不同矩阵描述，从实际运算性质来看并不是不分好环的。有些描述矩阵就比其他的矩阵性质好得多。这很容易理解，同一头猪的照片也有美丑之分嘛。所以矩阵的相似变换可以把一个比较丑的矩阵变成一个比较美的矩阵，而保证这两个矩阵都是描述了同一个线性变换。
这样一来，矩阵作为线性变换描述的一面，基本上说清楚了。但是，事情没有那么简单，或者说，线性代数还有比这更奇妙的性质，那就是，矩阵不仅可以作为线性变换的描述，而且可以作为一组基的描述。而作为变换的矩阵，不但可以把线性空间中的一个点给变换到另一个点去，而且也能够把线性空间中的一个坐标系（基）表换到另一个坐标系（基）去。而且，变换点与变换坐标系，具有异曲同工的效果。线性代数里最有趣的奥妙，就蕴含在其中。理解了这些内容，线性代数里很多定理和规则会变得更加清晰、直觉。
这个留在下一篇再写吧。
因为有别的事情要做，下一篇可能要过几天再写了
附注：bilibili有一份线性代数的教学视频，用图形化的方式很好的展现了矩阵的实际意义。
强烈推荐线性代数