I've just started a new term of uni and I just started a module in continuum mechanics, this being said there's some things that seem pretty important that I'm new to. The first exercise in the notes states to show various results regarding tensor product algebra, namely the first one being S(a ⊗ b) = (Sa) ⊗ b with a,b$\in{V}$ and S$\in{V^2}$ (We collect all vectors in the set V which forms a vector space over $\mathbb{R}$). Here's what I gather I can try to do to solve the equation from my notes:
As a,b$\in{V}$, we can write in a coordinate frame {e$^1$, e$^2$, e$^3$} that a=$\sum_{i=1}^3a_ie^i$ and b=$\sum_{i=1}^3b_ie^i$ and S can be written as a (3x3) matrix S$_{i,j}$, using this I assume we can separate out the equation S(a ⊗ b) = (Sa) ⊗ b, however our notes define the tensor product (a ⊗ b)v = (b · v)a ∀v∈V. I'm quite confused as to what the tensor product even is and how one applies it to two vectors, or how I can solve questions like this rigorously using a coordinate frame which I believe the exercise is asking me to do.