Month: June 2017

• Translated from: 只是一个记法
• Once in an interview, I was asked by the interviewer what the geometry meaning of cross product was. I stumbled in answering it as I am always reluctant to assign a meaning to it.

When I was learning linear algebra in my undergraduate years, my biggest confusion was about what interpretation I should assign to concepts like matrix, dot product, and cross product. After reading numerous books, there was an interpretation emerged in my brain vaguely, which I still couldn’t express in words. Failed to be articulated in words, this inner-understanding was good enough to unblock me to proceed to focused on more manipulations around those concepts (but still, with no precise interpretation I could attach to them).

Years later, when I began studying 3D rendering as a hobby, and frequently faced the task of drawing custom UI in my day job, these concepts gradually became a second nature, and I never bothered to seek their inherent interpretation. However, I often encounter beginners in this field asking the same questions I had in undergraduate years. I would bet that everyone who has been introduced to these concepts, whether they eventually pursued a career in graphics or drop the field after college lessons, has struggled with looking for an inherent interpretation. Reflecting on this, what shocks me the most is that textbooks, through generations, have never directly explained the meaning of these concepts. Hence, beginners have to grasp their meaning on their own. “The meaning of the dot product seems to be an area size, but the cross product seems utterly meaningless? It must be me not understanding deeply enough!”

In reality, the purpose of those concepts is similar to why a strict structure dictated ancient poem writing. Why did each line need to have a fixed number of characters with specific tonal patterns? Was it for aesthetics? While aesthetics played a role, if beauty was the ultimate goal, the constraints of the structure would have limited the poet’s expression. The true purpose of those rules was to facilitate easy memorization. The significance of structured poems was to ensure they could be propagated mouth by mouth before the invention of modern printing techniques. It was nothing but a notation.

Similarly, most concepts in linear algebra don’t have inherent meaning. They merely help scholars write formulas in a more beautiful way, making them easier to be memorized. This is akin to writing imaging formulas as 1/a + 1/b = 1/f. In the computer age, these mnemonics take on an additional significance. Due to their initial widespread application, special acceleration circuits were designed in hardware to compute them. There were also well-optimized software libraries. In addition to the easy memorization, the extra benefit in performance also encourages researchers to focus more on expressing theories in the form of these mnemonics.