The Philosophy of Linear Algebra
Whenever a student wonders why linear algebra, matrices, and vectors are important, the response from teachers usually sounds like this:
“Matrices are important because they are used everywhere, such as in smartphone technology, GPS, quantum mechanics, robotics, statistics, artificial intelligence, etc.”
This response is true, but I find it unsatisfying and incomplete. This line of reasoning is inductive, but I want something deductive. You could list hundreds of practical uses for linear algebra, but no matter how long you make that list you won’t be able to unravel the true nature of these mysterious number structures.
After thinking about and using linear algebra for several years, I have arrived at a couple ways to describe the true nature of linear algebra. In this article, I hope to explain the deeper reasons why it magically pops up in almost every corner of STEM.
Matrices are General
Before understanding why matrices are useful, we must first think about why numbers are useful at all.
Consider the number “5”. This single number can be used to describe any quantifiable entity that our imagination allows: 5 trees, 5 personalities, 5 doors, 5 planets, 5 electrons, 5 religions, etc. The range of ideas that can be expressed with numbers is what makes them powerful and useful.
However, there are some ideas that a single number cannot describe. Specifically, the range of ideas can be extended by putting together structured combinations of numbers, and this is how the matrix is born.
This method of extending a range of ideas can be easily compared to the Roman Alphabet. A random list of letters doesn’t mean much by itself, but a particular permutation of letters forms a word which has a meaning beyond what a random list of letters could provide. When you read a word, you are not processing every letter separately. You look at the group of letters and extract one meaning, a meaning that is not present in any of the individual letters making up that word.
A matrix operates on the same philosophy. A random list of numbers can be useful, but if you put them in a particular arrangement, such as an upper triangular matrix, a whole new story about the numbers is told. An upper triangular matrix conveys a whole lot more than just the fact that the numbers form a triangle; it tells you that the matrix is describing a nonsingular system that is solvable. The diagonal matrix describes a system with elements that only operate on themselves, and a symmetric matrix tells the story of a system that is bi-directional. Linear algebra is beautiful because the geometric shape of the matrices is deeply connected to its algebraic behavior.
Different arrangements of numbers in a matrix also yield different eigenvalues and eigenvectors. I often think that the personality of a matrix is contained in the eigenvalues. Whether a matrix describes an expansion, rotation, or an oscillation is encoded in the eigenvalues.
A pile of random numbers with no structure cannot convey as much personality or tell as interesting of a story as a group of numbers that are arranged in a structure. If a single number is able to convey the general idea of quantity, and the matrix conveys an even more complex and general idea of quantity, then it becomes more obvious as to why matrices are so useful in describing the universe.
Before I move on to the next section, some of you may be eager to point out how tensors, co-vectors, and abstract algebra extend even further. Those elements are fascinating too, but that is outside the scope of this article.
Linear Combinations are General
“Nature is that condition, that reality, which in appearance consists in life and death, or, in other words, in the composition and decomposition of all things.” -ʻAbdu’l-Bahá, Some Answered Questions
Composition and decomposition are general processes that can be used to describe many transformations that we observe in the universe. If the universe is made up of so many compositions and decompositions, then it makes perfect sense to deduce that a mathematical operator describing these transformations would be incredibly useful.
This is where linear combinations enter the picture, as well as its extension, the matrix multiplication. Linear combinations can effectively describe composition and decomposition because it is simply a sum of products. It is also a fundamental building block of linear algebra. The dot product is a linear combination. The determinant is a multi-linear combination. A matrix multiplication is a chain of dot products, which is a chain of linear combinations.
When you consider how general addition and multiplication (which is just more addition) are, it follows that a string of adds and multiplies can represent and approximate a large range of more complex operators. Linear combinations can be used to compute derivatives, integrals, convolutions, finite state machines, fourier transforms, and more.
Matrices existing by themselves are powerful, but with the help of linear combinations, there is a general rule that dictates how these matrices interact with one another. So a general structure (the matrix) combined with a general operator (matrix multiplication) gives you a field that can describe countless phenomena.
Beyond Practical Coincidence
Interestingly enough, linear algebra only exposed its deepest secrets to us in the 20th century, despite having existed for many centuries. While nowadays most people learn linear algebra through the perspective of vectors and transformations, the matrix and determinant are much older than the ideas of vectors in a multi-dimensional space. Computers certainly helped accelerate the development of linear algebra and vice-versa, but computers are not the fundamental reason linear algebra is so ubiquitous. The path to understanding linear algebra does seem pretty arbitrary, but linear algebra doesn’t exist simply because humanity managed to concoct a system that coincidentally finds usage in many seemingly unrelated subjects. Linear algebra is the result of our attempts at describing the universe as accurately as possible, and its general nature allows us to do that.
