Are you having a difficult time trying to understand orthogonalization? Well, you’ve come to the right place. In this article, you’ll learn all about this topic, specifically the Gram-Schmidt orthogonalization and how to use its process calculator. It may seem daunting at first, but you should have a better understanding of this simple algorithm by the end of this article.
Understanding What a Vector Is
In school, probably one of the very first topics you’ll discuss and learn about is the subject of velocity. Once you’ve familiarized yourself with the formula v = s / t, you begin creating different scenarios wherein a car or vehicle is going a certain direction parallel to the road. You draw arrows on the vehicle to show where they’re headed, and your teacher would call it “the velocity vector.” Even in subjects outside of mathematics, such as physics, for example, you’d still find similar arrows meant to indicate the direction of a force. These arrows can be used to describe a plethora of things, but one thing is for certain, the arrow is known as a vector.
When we talk about general mathematics, a vector is defined as an element of a vector space. But what is a vector space? This is a set of elements that have two operations that meet the expectations of some natural properties. These elements can range from permutations, sequences, and functions.
Cartesian Vector Spaces
One example of a vector space would be a Cartesian space. This space recognizes that a number is a one-dimensional vector space. The plane on which we draw or write pairs of numbers is considered as vector space as well. It doesn’t end there, though: the three-dimensional space of this world is considered a set of three real numbers.
What Is the Meaning of Orthogonal?
When you say orthogonal, it basically has a similar meaning to the word perpendicular. In other words, you could say that orthogonal is, somehow, connected to the 90-degree angle between objects. And if you really think about it, this definition makes sense because orthogonal vectors are lines that have a right angle between them within two and three-dimensional spaces.
In the case of trying to decipher if a one-dimensional space has orthogonal elements, you’ll need to use a new tool. The dot product, otherwise known as the scalar product, of two vectors v = (a₁, a₂, a₃,…, aₙ) and w = (b₁, b₂, b₃,…, bₙ) is the number v ⋅ w given by v ⋅ w = a₁*b₁ + a₂*b₂ + a₃*b₃ + … + aₙ*bₙ.
If you inspect closely, the dot product or scalar product is only a number that you can obtain by your everyday addition and multiplication. Now that you know about this tool, you’re more than ready to define orthogonal elements in whatever situation you find yourself in.
Explaining the Gram-Schmidt Orthogonalization Process
So, what is the Gram-Schmidt orthogonalization process all about? It’s basically an algorithm that takes a set of vectors and comes back with an orthonormal basis of the span of the taken vectors. Follow the steps below if you want to use this process:
- Take the vectors whose orthonormal basis you’d like to look for. These could be within v₁, v₂, v₃,…, vₙ.
- Consider u₁ = v₁ and set e₁ to be the normalization of u₁.
- Take u₂ to be the vector orthogonal to u₁. Then, make e₂ the normalization of u₂.
- Select u₃ so that u₁, u₂, and u₃ are orthogonal vectors. Set e₃ to be the normalization of u₃.
- Simply keep repeating this same process until you no longer have any vectors.
- Voila! The e’s that are non-zero are the orthonormal basis.
Using the Gram-Schmidt Calculator
Imagine that there’s a game you enjoy playing, but it requires you to leave your house and go outside to enjoy it to its full capacity. However, you get badly injured one day, which means you can no longer walk around your neighborhood playing the game as much as you’d like. Thankfully, you have a friend who was able to help you out by discovering a program you could download into your phone; this program allowed you to virtually traverse the game’s world without having to leave your home.
It sounds like a game-changer, right? But here’s the catch: you need to input the vectors to determine where your virtual character could move in-game. Since we live in a 3D world, you have to use three-dimensional vectors, too. Not only that, the plugin requires the vectors to be orthogonal! You may be on the verge of giving up, but not all hope is lost just yet, thanks to the Gram-Schmidt calculator.
Since there are three vectors and each one has three coordinates, we start by letting the calculator know the specifics of the problem by selecting the right options, which can be found under ‘Number of vectors’ and ‘Number of coordinates.’ After inputting all the necessary numbers, the Gram-Schmidt calculator will easily give you the answer. This saves you the trouble of having to manually find the orthogonal basis every time.
There you have it—a brief overview of the Gram-Schmidt process. We hope you were able to learn something, and if things get too difficult, know that you could always rely on the orthogonal basis calculator to spit out answers.