Unlock the Power of Basis Vectors: Impactful Examples

[matqkks]

I have normally introduced basis vectors by just stating independent vectors that span the space. This is perhaps not very inspirational.
What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. Maybe a good example will do.

 

[nonequilibrium]

Perhaps that the idea that each basis vector introduces a new direction in "space", this captures simultaneously (in an intuitive sense) both the idea of "independent" (i.e. the directions are independent, meaning no matter how much you move in one of the other directions, you won't end up in that specific direction, and if you have all directions (the number coinciding with the intuitive dimensionality of the space) then you can't add a new direction that is independent) and the idea of "spanning".

Simplified: a basis vector symbolizes the notion of "new direction".

Does this help? Or looking for something else?

 

[chiro]

It's probably a good thing to outline things in terms of decomposition.

Outline that a lot of what any kind of analysis is about is breaking things down into separate components. Independence means things are separate in a sense. Orthogonal means things are separated in a way that every component is completely distrinct and separate from the other.

Independence is a way of clarifying of this difference and orthogonality is the rigorous of way of saying that two things are completely independent of each other: the intuition is if I change one thing that is completely orthogonal to another, I don't change the other thing at all.

This plays into dimension which basically finds the minimum number of orthogonal components which is the minimum description of the system (we are dealing with linear systems).

This then gives us the simplest description of a system (i.e. reduces it to a minimal form) which is useful for understanding the system because it can not be reduced further.

 

[genericusrnme]

I second what chiro said

you could also show that functions can be vectors, I found that pretty cool when I first learned about it, maybe show them that the polynomials of degree less than n is a vectorspace (of course, don't give this out as your first example)

 

[Matrix0]

Basis vectors are the building blocks of linear algebra, and unlocking their power can lead to a deeper understanding of mathematical concepts. Rather than simply stating that basis vectors are independent vectors that span a space, let's explore a more attractive and impactful way to introduce them.

Imagine a group of dancers on a stage, each moving in their own unique way. Individually, they may seem chaotic and disconnected. But when the choreographer gives them a set of instructions, suddenly their movements become synchronized and purposeful. In this analogy, the dancers represent the basis vectors and the choreographer represents the linear transformation that brings them together.

Now, let's apply this concept to a real-world example. Consider a map of a city, with each street representing a different basis vector. Individually, these streets may seem disjointed and confusing. But when we use a set of directions (a linear transformation), suddenly we can navigate through the city with ease and efficiency. This is the power of basis vectors – they provide a framework for understanding and navigating complex systems.

By introducing basis vectors through relatable examples, we can ignite a sense of curiosity and motivation in our students. They will be inspired to see how these seemingly simple vectors can have a profound impact on understanding and solving complex problems. So, let's unlock the power of basis vectors and empower our students to think critically and creatively about the world around them.