Turning sum notations (see below) into a compact vector notation is always hard work for me. I was wondering if there are any rules where you can directly go from one well-formed sum to a vector notation. For example, I can't figure out how to express the below sum into a vector expression:
Let $f(\textbf{y}) = (\textbf{x} - \textbf{Vy})^T(\textbf{x} - \textbf{Vy})$, where $\textbf{x}$ is an $m$ dimensional vector, $\textbf{y}$ is an $n$ dimensional vector, and $V$ is a $m$x$n$ dimensional matrix.
I am attempting to turn the term $\partial{f}/\partial{y_c}$ below into vector notation.
$$f(y) = (x-Vy)^T(x-Vy)=\|x-Vy\|^2=\|Vy-x\|^2$$
$$\nabla_yf(y) =2V^T(Vy-x)$$
If you want to extract the $c$-th component, just multiply it with $e_c^T$ where $e_c$ is the unit vector with the $c$-th entry being $1$ and the other entries $0$.
$$\frac{\partial f}{\partial y_c}=2e_c^TV^T(Vy-x)$$
In general, some useful identities are
$$\sum_{i=1}^n x_iy_i = x^Ty$$ $$\sum_{i=1}^n x_i^2 = x^Tx=\|x\|^2$$
and being familiar with matrix calculus helps.