Is sampling from a transformed pdf the same as transforming samples in the same way from the original pdf? Suppose I have a pdf $f(\vec{v})$ over $\vec{v}$ and define $\vec{u}=T(\vec{v})$, i.e. a transformation of $\vec{v}$. Would sampling $f(\vec{u})$ be the same as taking samples $\vec{v}_i$ from $f(\vec{v})$ and transforming them according to $T(\cdot)$? The answer intuitively feels like it should be yes, but it would be nice to have a more rigorous way to show this.
Would this still apply if $T(\cdot)$ is not bijective, i.e. if $\vec{v}$ has more elements than $\vec{u}$? As a concrete example, suppose
$$
\vec{u}=T(\vec{v})=\begin{bmatrix}
2 & 0 & 1 & 4 \\
0 & 3 & 0 & 2 \\
\end{bmatrix}\vec{v}.
$$
Is there a sampling method for which this transformation of samples is a basis?

Yes, these are the same things. It's a fact frequently exploited in MCMC methods for Bayesian models (see e.g. Section 1.9 of the 3rd edition of Bayesian Data Analysis).
