We were given this identity during the lecture, but the proof was omitted:
$$\langle v, u \rangle u = (uu^T)v \text{ with } ||u||=1$$ If I write out the intermediate matrices, I see that the equality holds, but I wanted to prove it more analytically.
I tried starting with $\langle v, u \rangle u = (v^Tu)u$ using the definition of dot product, then the professor suggested we use associativity of matrix/vector multiplication to rewrite the operations as an outer product.
I'm not very familiar with the outer product and I do not get how to do use it for the last part. Any insight appreciated.
The inner product is scalar valued, so can be placed on the right of $u$ instead of on the left. We can also write it as $u^Tv$ instead, because the product is symmetric. The associativity of matrix multiplication completes the proof.