= Linear left inverse criterion
{title2=$ST=I_V\iff\ker T=\{0\}$}
A <linear map> $T:V\to W$ between <finite-dimensional vector spaces> admits a linear <left inverse> $S:W\to V$ if and only if its <kernel of a linear map> is $\{0\}$, equivalently if and only if it is <injective>.
Necessity follows from $Tv=0\Longrightarrow v=STv=0$. For sufficiency, choose a <basis> $v_1,\ldots,v_n$ of $V$. Triviality of the <kernel of a linear map> implies that $Tv_1,\ldots,Tv_n$ are <linearly independent>. By <basis extension>, they extend to a <basis> $Tv_1,\ldots,Tv_n,w_{n+1},\ldots,w_m$ of $W$. Define $S(Tv_i)=v_i$ and $S(w_j)=0$, and extend by <linearity>. Then $STv_i=v_i$ on a <basis>, proving $ST=I_V$.
The freedom in $S$ lies in its values on a <direct-sum complement> of the <image of a linear map>. For example, for $T(x,y)=(x,y,0)$, every <linear map> $S(u,v,w)=(u+aw,v+bw)$ is a <left inverse>, with arbitrary <scalars> $a,b$. A <right inverse> instead requires $T$ to be a <surjective linear map>.
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