This paper investigates the problem of identifiability and estimability of linear structures in $n$ dimensions. The concept of identifiability is examined to elucidate the senses in which it may be interpreted in the present problem. Particular attention is given to the question of treating linear subspaces rather than specific coordinate systems. Necessary and sufficient conditions for identifiability are obtained under the assumption that the "errors" follow a multinormal distribution.
T. A. Jeeves. "Identification and Estimation of Linear Manifolds in $n$-Dimensions." Ann. Math. Statist. 25 (4) 714 - 723, December, 1954. https://doi.org/10.1214/aoms/1177728657