If you run out of time, this lecture can be summarized in a few minutes by telling students that they know about 2-d and 3-d arrows in space. These vectors form what is formally called a vector space. The key features of a vector space is that the set of objects can be “added” and “multiplied by numbers (scalars)” and that the resulting objects are also vectors in the vector space. This property is called “closure”. There are many types of vector spaces which they will be learning about in their future physics courses. Make sure to follow up when they first learn about Fourier series.

Somewhere in their junior year, students should be introduced to the formal properties of a vector space.

1. Lecture/Discussion about the properties of linear vector spaces:
   • SWBQ: in groups, talk about the properties of vectors (e.g. 2D arrows) and write one on each small whiteboard.
     • Use responses to generate a list of properties of vectors.
   • SWBQ: draw a function that has a period of $2\pi$. (This turns out to be a great SWBQ! Emphasize that $\sin\theta$ and $\cos\theta$ or even $\sin(2\theta)$ are not the only such functions. Also emphasize that if you only draw the function between $0$ and $2\pi$, you don't know whether or not it is periodic.
     • This vector space can be used as an example to discuss how concepts like length, direction, dimension, addition, etc. are generalized.
2. Give a pretty standard lecture stating the properties of abstract vector spaces (see for example (RHB 8.1). Use bra/ket notation and generalize the results of the SWBQ's above to the idea of abstract kets. Emphasize:
   • A vector space requires a definition of “addition”. What is that definition for the vector spaces of arrows and of functions with period $2\pi$?
   • A vector space requires a definition of “scalar multiplication”. What is that definition for the vector spaces of arrows and of functions with period $2\pi$? This can be a good place to introduce the idea that the scalars for some vector space are real numbers but for other vector spaces they can be complex numbers. When they are complex numbers, we may lose the ability to represent the vectors in a drawing.
   • The two properties above are what allow you to choose a basis and expand any vector as a linear combination of a basis. If your student have already seen Fourier series, this is the place to emphasize that $\sin\theta$ and $\cos\theta$ and $\sin(2\theta)$, etc. are the basis vectors for the vector space of functions with period $2\pi$. The fourier coefficients are just the coefficients of the “vector” in that expansion.
   • You can touch only lightly on the other properties (commutativity and associativity of various kinds, zero vector, inverses).
3. Go on to give a pretty standard (short) lecture on the properties of inner products (see RHB 8.1.2), again using the two examples of arrows in space and periodic functions as examples.