誰能說說矩陣的乘法幾何意義，越通俗越好

作者：毛毛吉吉

鏈接：/question/28623194/answer/104420761

來源：知乎

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There is my understanding. The basic idea of the matrix with rows and columns is likely a X-Y axis which means a 2-dimensional space. So that we consider row and column into two parts generously. If you are more interested in the row relation, (such as X-axis you're interested) you will get your point of view of the problem from a X-axis' perspective. In another word, you can imagine you are just standing on some position at X-axis from original point to positive X-axis (NOTICE: X-axis is a 1-dimensional space), and you're more willing to tackle your problems using the solutions in 1-dimension. Don't worry! Let me describe it more vividly.

The problem is lying in a 2-dimensional space, and you want to solve it using an approach of 1-dimensional perspective. Why should we do it (Question 1)? How can we do it (Question 2)? The answer to the Question 1 is that it is more easy than a 2-dimensional problem to solve from our experiences and conclusions in the most time. The answer to the Question 2 is more complex. Matrix is a smart way to compress a 2-dimensional question into a 1-dimensional question. The approach is, of course, differentiating a row and a column. You can continue imagining you're standing at the X-axis with a 1-dimensional point of view being ready to tackle a problem you faced. You are succeed ignoring the Y-axis, no matter what happened on it, because you have no ability to meet anything in the second dimension (Y-axis). That's a simple way for a problem solver.

Hold on Question 2 :)

Imagine a compressor compress a square biscuit at one direction, referencing Figure 1.

You're more interested in X-axis, and you don't care what append on Y-axis. All right, actually, Some solvers think they firstly consider each column contains homogeneous elements and the column could be compressed. Later, we use Compressing and Decreasing Dimension Method (I named it CDDM). Reference Figure 2.

For this reason, you are more likely walking on the X-axis form point O(0,0) to point A(0,a) where ‘a’ is a real number on X-axis, and considering a 1-dimensional problem instead. You will see how we replaced and simplified the question, referencing Figure 3.

Do you got the Compressing and Decreasing Dimension Method (CDDM)? Let me draw a simple conclusion. The matrix is a 2-dimensional problem. We use CDDM to simplify it, that is, we chose row or column to calculate and proof a theorem or problem. This approach is likely decreasing the dimension, I think. And the rule is we believe each row or column has the coordinating and corresponding properties for every elements contained by a row or column.

Yes, matrix is a container! If you're more interested in a row relation, and imagining walking on a X-axis, you will believe there is no column so that you compress the columns into single elements. And then, you walk from original point O(0,0) to point A(0,a), where ‘a’ is a real number on X-axis. It's more easy to find out the row relation in the matrix, isn't it?

Now, I suggest you think again about the form of the matrix below in Figure 4. Why do we write it like that?

I hope I could explain the Question 2 more clearly. But, you know, English, as a second language is not too fluent for me. What I want to highlight at last is that row and column is the same, which is the major relationship between rows and columns, I think. The only differences are the angle we tackle a problem and the way we understand a knowledge. CDDM is a useful attitude in our real life.