= Full column rank
{title2=$\operatorname{rank}A=n$}
An $m\times n$ <matrix> has <full column rank> when its <matrix rank> is $n$, equivalently when its columns are <linearly independent vectors>. This requires $n\leq m$. Its <kernel> is $\{0\}$, so the represented <linear map> is <injective> and every solvable <linear system> $Ax=b$ has a unique solution. It need not reach every output.<Interactive Linear Algebra by Dan Margalit and others/One-to-one and Onto Transformations>[Interactive Linear Algebra, §3.2.1, One-to-one matrix transformations theorem and Wide matrices do not have one-to-one transformations]{ref}
For example,
$$
B=\begin{pmatrix}1&0\\0&1\\1&1\end{pmatrix}
$$
has <full column rank> two, but its <column space> is the <plane> $b_3=b_1+b_2$ in $\mathbb R^3$. An attainable output fixes the input uniquely as $(b_1,b_2)^T$; an output outside this <plane> has no preimage.
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