Provided the MOFs are distinct, the two rows of J are highly unlikely to point in exactly the same direction; thus, we safely assume J is fullrank, i.e., σ 2 > 0. Eqns. 10 and 11 indicate the sensor is responsive to gas composition changes in the direction of v 1 and v 2 but more sensitive to changes in the direction of v 1 (σ 1 ≥ σ 2 by definition, but likely σ 1 > σ 2 ). Eqn. 12 indicates the null space of the Jacobian matrix is non-trivial and spanned by v 3 . Therefore, the linear map ã : x → m is non-injective, as a consequence of M < G. Consider small gas composition changes (of magnitude > 0) from x op . The linear approximant ã(x) in eqn. 2 maps a sphere in 3D composition space centered at x op to a solid ellipse in 2D sensor response space centered at m op . See Fig. . This follows from eqn. 2 and eqns. 10-12. The composition change in the sensitive direction, v 1 (green in Fig. ), elicits the largest sensor response-from m op to m op + σ 1 u 1 on the endpoint of the major axis of the ellipse. The sensor is responsive, but less sensitive, to composition changes in the direction of v 2 (yellow in Fig. ); the composition change v 2 moves the sensor response by the vector σ 2 u 2 , placing it on the endpoint of the minor axis of the ellipse. The relative sensitivity of the response to composition changes in the direction of v 1 vs. v 2 depends on the singular values, σ 1 vs. σ 2 . On the other hand, v 3 is the unresponsive direction; the composition change v 3 (red in Fig. ) does not elicit a sensor response, and m remains at m op despite the change in gas composition. Using this toy sensor array, we therefore cannot detect small composition changes in the direction of v 3 ! 5. Demonstration: natural gas sensing with gravimetric sensors based on HKUST-1 and Co-MOF-74