Consider the vector $\mathbf{x} \in U$, where $U = \mathbb{C}^3$ $$\mathbf{x} = \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}$$
$U$ has a basis of three vectors: $$B_U = \left\{\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right\}$$
Now consider multiplying $\mathbf{x}$ by a $3 \times 3$ matrix $\mathbf{A}$ of rank 2 to yield $\mathbf{y}$:
$$\mathbf{y} = \mathbf{A}\mathbf{x}, \qquad \text{where $\mathbf{A}=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}$}$$
$$\implies\mathbf{y} = \begin{bmatrix}x_1 \\ x_2 \\ 0\end{bmatrix}$$
The result of this multiplication is that $\mathbf{y}$ has no components corresponding to basis vector $[0,0,1]^T$, and so $\mathbf{y} \in V$ where $V$ is the subspace of $U$ that has a basis of only two vectors:
$$B_V = \left\{\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}\right\}$$
The process of multiplying $\mathbf{x}$ by $\mathbf{A}$ can be referred to as the projection of $\mathbf{x} \in U$ onto subspace $V$. $\mathbf{A}$ is a projection matrix and it possesses the defining property that $\mathbf{A}^2 = \mathbf{A}$.
Now let's consider multiplying $\mathbf{y}$ by a rectangular matrix $\mathbf{B}$ to yield $\mathbf{z}$:
$$\mathbf{z} = \mathbf{B}\mathbf{y}, \qquad \text{where $\mathbf{B}=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$}$$
$$\implies\mathbf{z} = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$$
This results in $\mathbf{z} \in W$, where $W = \mathbb{C}^2$.
$W$ has a basis of two vectors:
$$B_W = \left\{\begin{bmatrix}1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1\end{bmatrix}\right\}$$
Therefore, even though $\mathbf{B}$ is a rectangular matrix, there is a one-to-one correspondence between the elements of $V$ and the elements of $W$ via multiplication by $\mathbf{B}$. In other words, it allows the three elements of $\mathbf{y}$ to be reduced to just the two elements of $\mathbf{z}$ without loss of information. In fact, in the context of transforming between vector spaces $V$ and $W$, there exists an inverse $\mathbf{B}^{-1}$ which allows recovery of $\mathbf{y}$ from $\mathbf{z}$:
$$\mathbf{B}^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{bmatrix}$$
So my question is...
What is the name to the operation performed by multiplying by $\mathbf{B}$? That is, what is the name of the operation that takes a vector and turns it into a vector of lower number of elements without loss of information? I think isomorphism might be a close term, but it might not seem to suggest the notion of reducing the number of elements of the vector.
(A further bonus question: is there a name to the operation of multiplying $\mathbf{x}$ by $\mathbf{A}\mathbf{B}$? This appears to behave like a projection operation, except that the output has a lower number of elements than what would arise in a projection like $\mathbf{A}$)