How Can Basis Vectors Simplify Real-World Problems?

[matqkks]

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. I have normally introduced it by just stating independent vectors that span the space.

 

[Evgeny.Makarov]

I've never taught linear algebra, but selecting a basis is essentially selecting a coordinate system, which allows identifying each point (vector) with a unique sequence of numbers. Coordinate systems are obviously crucial for solving many practical problems.