July 14

Explaining Basis Vectors

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If you’re interested in knowing all about Deep Learning or perhaps even Machine Learning, then it’s fundamental that you learn and understand Linear Algebra because the principles and methodologies taught here are crucial when it comes to those kinds of concepts.

With Linear Algebra, you have the opportunity to concisely define and perform operations on higher-dimensional coordinates. If you’re familiar with this branch of mathematics, you most likely know that linear equation systems is the main focus.

As such, learning about Linear Algebra wouldn’t be complete if you also didn’t learn what basis vectors are. Fortunately, this article will help you do so; you can expect to find all kinds of information around basis vectors, including the concept behind them, their properties, and more.

Defining Basis Vectors

Before anything else, let’s start by defining what basis vectors are. Consider this: if you’re able to write down every possible vector in a given space as a linear combination of a few vectors, and then these vectors happen to be independent of one another, then these vectors are considered as basis vectors for that provided space.

The Idea Behind Basis Vectors

So, what is the concept behind these basis vectors? Let’s use an R-squared space as an example. This space basically means that we’re taking a look at vectors within 2 dimensions. This also means that each of these vectors contains 2 components. With that in mind, there should be an infinite number of vectors overall, which will be found within 2 dimensions.

Basically, this also means that we’re capable of representing these different vectors by just using basic elements and then using a combination of these elements.

What Properties Do Basis Vectors Have?

How do you know you’re dealing with basis vectors? As you may or may not know, basis vectors actually have some basic properties. If you’re having trouble deciphering whether a particular vector is a basis vector or not, refer to these different properties to help you out.

They Need to Be Linearly Independent

Consider this: if v1 is multiplied by any scalar, it’s impossible to find the vector v2. What does this mean? This leads us to the conclusion that both v1 and v2 are linearly independent from one another.

This is a good thing if you’re looking for basis vectors because you want basis vectors to be linearly independent. This is because we want these vectors to produce unique information or data. If these vectors aren’t independent, then this means they won’t be able to offer anything unique.

They’re Not Unique

While we want basis vectors to offer unique information, this doesn’t necessarily mean that basis vectors themselves are unique. In fact, you can find different sets of basis vectors, and there are several ways you can define them too. Nevertheless, all of them should have the same properties that make them basis vectors.

They Need to Span the Entire Space

Last but not least, it’s a requirement for basis vectors to span the whole space. In this case, “span” means that any vector found in that particular space can be written as a linear combination of basis vectors.

Conclusion

Don’t be afraid to keep learning, even if you think that a certain concept seems too difficult to grasp. If you’re interested in learning about data science, then you need to branch out your knowledge and master Linear Algebra too. It may seem daunting at first, but with enough practice, you’ll soon discover that these concepts are just a piece of cake.


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algebra, mathematics


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