Supposed I have 2 vectors $a = (a_1, a_2)^T$ and $v = (v_1, v_2)^T$ (column vectors) with $||a|| = \sqrt{a_1^{2} + a_2^{2}}$ and $a, v \in R^{1\text{x}2}$. Let $A = ||a||^2v$, and:
$$A = ||a||^2v = (a^Ta)v = va^Ta$$
I do not get the part from $(a^Ta)v = va^Ta. (1)$ If I think about their dimensions, it is supposed to be (1x2 2x1)(2x1) = 2x1 1x2 2x1, which is correct in this case. However, I wonder if there any multiplication rule that implies equation $(1)$?
The multiplication rule used is the following: $$\forall v \in \mathbb{R}^{n \times m}, \forall \lambda \in \mathbb{R}, \lambda v=v \lambda$$ Indeed, we know that $a^Ta$ is a real, positive number (since it is equal to $||a||^2$). Therefore, you can write $\lambda = a^Ta$ and use the previous rule.