Assume we have a matrix
$$A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} $$
and a vector $x$
$$x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} $$
And a transformation $T_A$ is defined as $T_A(x)=Ax$
The first task is to find the domain and target space of the transformation.
By the rules of matrix-vector multiplication,
$$ Ax = \begin{pmatrix} x_1 \\ 0 \end{pmatrix} $$
Does it mean that the domain is $\mathbb R^2$ by the number of columns in A, but the target space is $\mathbb R^1$ as we've got only $x_1$ in the end?
Best Answer
The domain is $\mathbb{R}^2 $ because are considering $T$ as
$$T:\mathbb{R}^2\rightarrow \mathbb{R}^2$$
But the transformation is not surjective, and you can produce the range of $T$ with $$ \begin{pmatrix} 1 \\ 0 \end{pmatrix} $$
So the range is one dimensional, that is $\mathbb{R}^1.$