The latest version of this article is located at: http://kexue.fm/archives/2208/
Dear readers, the "Understanding Matrices" series on Scientific Space has reached what I consider to be the most exciting part: the description of determinants. This part was not discussed in Meng Yan’s articles; it consists of some perspectives I have derived by combining various books and online resources. The primary book referenced is The Mathematical Bridge. Tracing back to the source, what prompted my research into this area was the article "How Textbooks Should Be Written" by matrix67. This article contains a significant amount of intuitive understanding. In my view, while these contents might not represent the "orthodox" viewpoint, they can, to some extent, facilitate our understanding of linear algebra.
Most linear algebra textbooks introduce determinants through systems of linear equations. This approach allows students to quickly master the calculations and provides a practical application (solving systems of equations). However, it easily leads readers into a misconception, making them think that linear algebra is merely the study of solving equations. This does not allow readers to truly grasp its essence. Only when we have an intuitive and proficient feel for it can we apply it effectively.
The emergence of the determinant is actually to determine whether a matrix is invertible. It achieves this by constructing a "relatively simple" function, which is the determinant of the matrix. Let us reflect: what does it mean for a matrix to be invertible? As mentioned previously, a matrix is a transformation from one point to another; thus, the inverse matrix is clearly intended to transform it back. We also said that "motion is relative," and the transformation of points can be realized through the transformation of coordinate systems. However, according to our intuition, different coordinate systems should not have any differences other than computational complexity (for example, calculating dot products in a general affine coordinate system is more complex than in a Cartesian system). In physical terms, all coordinate systems are equal. Given one coordinate system, one should naturally be able to transform to another and naturally transform back. Since a matrix is a description of such a coordinate system, the only possibility for a matrix to be non-invertible is:
The n column vectors of this n-th order matrix simply do not constitute a coordinate system for an n-dimensional space.
In other words, it is not qualified to be a coordinate system! (In technical terms, these n vectors are linearly dependent.)
Therefore, we try to construct a function |A| for an n-th order matrix A such that when |A| is 0, it means the matrix is not qualified to be a coordinate system, implying it is non-invertible.
Let A = [\vec{c_1}, \vec{c_2}, \dots, \vec{c_n}], where \vec{v_i} (or \vec{c_i}) are the individual column vectors. Let’s think about which cases clearly disqualify matrix A from being a coordinate system:
Obviously, whether it can form a coordinate system is closely related to every vector \vec{v_i}. Thus, |A| must be a function of every vector \vec{v_i} to satisfy the requirement.
For simplicity, it is best if our function is linear. That is, if A = [\vec{c_1}, \dots, \vec{v_i}, \dots, \vec{c_n}] and A' = [\vec{c_1}, \dots, \alpha \vec{v_i}, \dots, \vec{c_n}], then |A'| = \alpha |A|.
Forming an n-dimensional coordinate system requires n independent vectors. If two of the n vectors in matrix A are identical, it naturally cannot form a coordinate system, meaning |A| = 0 in this case.
To determine an appropriate scaling factor and to satisfy our intuition, we define |I| = 1.
Surprisingly, based solely on the above properties, one can determine a unique function |A|, which is the determinant calculation formula provided in our textbooks! Of course, the derivation process is somewhat complex and will not be presented here. Interested friends can perform calculations with second-order and third-order matrices and then use analogy.
Let us recall how textbooks introduce determinants: first, the calculation formula for determinants is given via systems of linear equations, and then various properties of determinants are proven, such as "multiplying one row by 5 multiplies the value of the determinant by 5," and so on. Compared to the explanation above (which is actually the standard way of teaching linear algebra), this feels a bit like putting the cart before the horse. Since the determinant itself is defined by those properties, why should we use the calculation formula to reverse-engineer those properties? Other than exercising calculation skills, I really don’t see much purpose in it.
The content above also has a very intuitive geometric interpretation:
What is the necessary and sufficient condition for three vectors in three-dimensional space to form an affine coordinate system? It’s simple: these three vectors must not be coplanar! Therefore, generalizing to n-dimensional space, it is obvious that:
n vectors forming a coordinate system means these vectors are not coplanar!
Of course, a "plane" in n-dimensional space does not refer to a two-dimensional plane, but to an (n-1)-dimensional object, which is relative to n-dimensional volume.
It can also be stated in another way:
n vectors forming a coordinate system means the volume of the n-dimensional parallel solid (think of our parallelepiped) formed by these vectors is not zero!
Note that this involves volume. Whether it is zero or not relates to the invertibility of the matrix, which is exactly the same as the determinant. Is this n-dimensional volume related to the determinant? Let’s analyze some properties of the volume V of the n-dimensional solid formed by matrix A:
Clearly, each vector \vec{v_i} is also an edge length, so V should be a function of each vector \vec{v_i}.
When a certain vector becomes \alpha times its original size, the volume should also become |\alpha| times the original. The absolute value is added because vectors can extend in the opposite direction, but volume must be positive.
When two vectors are the same, it means these vectors are "coplanar," and the volume is naturally 0.
The volume of a unit n-dimensional cube is naturally 1.
Surprisingly, these properties correspond almost one-to-one with the determinant, except for the absolute value required for \alpha in point 2. Thus, these properties ultimately force the volume of this n-dimensional solid to be V = \text{ABS}(|A|), where I use ABS to denote the absolute value.
This is truly a wonderful fact. The determinant has such a clear geometric meaning: it can be used to find volume (in the case of a second-order matrix, it is the area of a parallelogram)! This explains why the parallelepiped volume formula we first learned in analytic geometry looks the way it does. It can be derived independently without relying on dot products or cross products! There are many other examples that readers will surely encounter when learning related knowledge. But on the other hand, this also reflects the poor quality of our textbooks: if we used the way we are taught determinants, how much effort would it take to prove that the determinant is volume?
These are my humble views; I welcome everyone to point out any errors.
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