Househbolder transformation identity matrix dimensions

Question

When performing a householder transformation and generating an elementary reflector matrix of the form:

$$H = I - 2\dfrac{vv^T}{v^Tv}$$

How do we know the dimensions of the identity matrix?

Answer 1

Let $n\in \mathbb N$ such that $v \in \mathbb R^n$. Then $vv^t \in \mathbb R^{n\times n}$ and we therefore need $I \in \mathbb R^{n\times n}$ for the difference to make sense.