I am trying to write a hook for vectors on a linear algebra course. Does anyone have an opening hook for a section on vectors that will have a real impact on students?
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I would start with the physical interpretation of vectors first, since it's the easiest to grasp, and easy to see the use of. Go through a physically motivated example, emphasize important points (like a basis, or linearity), and present a different example that might motivate you to generalize the definition (like the basis of a function space). Beginner linear algebra students can sometimes balk at the abstraction, since they're often used to more concrete ideas and an emphasis on calculation. In general I would do
physical example $\rightarrow$ calculation $\rightarrow$ abstraction $\rightarrow$ new example $\rightarrow$ sexy answer (to an otherwise awful problem)
The major thing is that the students see a need for linear algebra. If they're happy with their knowledge of vectors so far, why would they care about abstracting and looking at certain properties of them? You should be giving the idea of a bridge that lets you carry tools from one area (vectors here) into all sorts of other applications that don't look at all like the original, and that you need to know your bridge is sturdy before you start carrying things across.
My particular favourite example is the fourier transform and signal processing. You don't have to get into the calculus specifics, but it was a great moment for me when I realized the underlying idea of basis functions, and that not all vectors look like vectors.
I think the use of linear algebra for clasification of web pages by Google is a very interesting and actual hook.