Matrices — Mathematics of Perceptions

An educational article on Towards AI reinterprets matrices as tools for translating between different perceptions of space, using a story about giving directions to explain vectors and spanning vectors. The piece argues that matrices encode both direction and magnitude, serving as a shared framework to reconcile individual perspectives.

Last Updated on August 24, 2026 by Editorial Team Author s : Wuiii Originally published on Towards AI. Matrices — Mathematics of Perceptions Note: This article assumes you have read about matrices but didn’t understand its geometric meaning or significance or didn’t intuite about it. The article helps understand while revisiting your already read concepts. Spoiler alert This is a very interesting topic. Perhaps, the most interesting fundamental thing in that is the beauty in thinking and intuiting of such great ideas. My approach while unvieling the beauty is that, let us first consider there is no such thing as matrix. Then, what we do is we will slowly re- discover it. I hope this will be an interesting way. Consider that, you are calling your friend Rene to get apples from your another friend Isaac. Rene asks you Where is Isaac? You respond with something like: Well, you can find him if you go rightwards. Rene is obedient he listens to you and goes to his rightwards and doesn’t find him. He instead misses him. You, Isaac, Rene Map As we can totally see that there is a problem in our general approach to just tell something is right or left or any direction. This problem can be thought as the way you percieve the word and its routes is different from the way Rene percieves. And these different perceptions over the same thing is causing problems. Earlier, or the most normal way we solve this problem is let them both agree on a frame of reference. Let us give some framework called North-West-East-South to each of them. And, whenever they speak, they must follow the framework. In this way, they can work together. Well, this is definetly a solution and that is the general thing we do. How do we represent this framework and explain it to them for practical use? What exactly is North/West/East/South to each of them? Let’s say we tell each of them what are these directions in their own perspective. This solves the problem in different perceptions of directions for each of them. Now, they can follow the commonly thought framework to know directions. Lets now continue our story, now again on another day you call Rene and tell to meet Isaac to get apple. Now, you clearly mention Rene to go north to find him. Rene thinks that, its just a few minutes of ride as told by you and he goes towards north. But, in realty he feels that he traveled quite a lot than you told it took him an hour almost. And as you investigate about this you realize that the distance with which you and Rene is traveling in a unit time is different to each other. So, now you have to even make some standardized framework about how much is one step distance in unit time assuming a constant speed. Concluding from what we have learnt. We had require direction and also magnitude of each step in our framework. Does this sound familiar? Quantities that have magnitude and direction are called vectors. So, a better solution for this problem would be vector based framework. In other words, we would use vectors to somehow solve this. This is an intuitive idea before proceeding to the next section. From our understanding till now, if you want to tell Rene about the position of Isaac, you need a ordered collection of information. The ordered collection Lets call it Ord. Collection contains two very important vectors. These vectors act as translators between the shared framework and Rene. The first vector tells What direction is east? according to Rene’s perspective. While, the second vector tells what direction is north according to Rene’s perspective. Recall that again, vector doesn’t just say about direction, it also says about step length/distance per unit time . These ord. collection of vectors which we are talking about are called spanning vectors. Why? Because they teach us how to span/measure a perception of space. Now, lets rethink the whole thing so that we get a clear idea on this. In our example, we had two problems. Firstly, people have their own perception of space so their coordinate plane or way of measuring and the axes itself might be in different direction. The other possibility is their one unit of distance is different that the one unit of distance according to others perception. To solve this, we had given each of them ord. collection of vectors to get to a shared framework . These vectors tell how much one unit is in each axis by their magnitude and what direction the axis is. Now, each point in space, can be said as a combination of these vectors. We could say something like these many X axis unit vectors and these many Y axis unit vectors gives this point. This is why I say that these vectors are kinda rules which defines a perception of space. Example: A perception of space would be something like a gaint space where our 1 unit can be actually 10 units in normal cartesian space. Or a space in which the axis itself are tilted. A ordered sub-collection Lets call it, ord. sub-collection of those spanning vectors are called basis vectors. Basis vectors are a little bit special vectors. Why? Because they are those spanning vectors which are unique in a way that the other spanning vectors even if scaled and combined in various proportions can’t make up these vectors. In this section we will study how we represent the above ideas more mathematically and try to guess why do we do that in a certain way. Starting with vectors, lets consider our general notation inspired by our physics classes on how we refer to a point 3,4 using a position vector as described below. Eq. 1: Exemplary vector representation in high school physics Suppose, we have huge dimentions. How should we represent now? Example a point 2, 3, 5, 2, 1, 3 or something like 4, 2, 7, …