Why Are Some System States Indistinguishable in Observability Analysis?

[pj33]

Homework Statement

Why the null space of the Observable matrix is not the same as the state vector?

Relevant Equations

None

https://en.wikipedia.org/wiki/Observability

I am studying observability and I try to get some intuition on the topic.
Given the observable matrix, we can find the null space. However, the vectors of the null space are states but this differs from the definition of what a state vector of a system is. In addition, the null space indicates that all the states with the same velocity are indistinguishable. Why is this the case ?

Why is that?
Is it because the state vector is a vector containing other vectors (position velocity etc), that's why it differs from the state in the null space.

 

[Valkarie]

The state vector of a system usually contains information about the position, velocity, and other physical properties of the system. The null space of a matrix contains vectors that can not be expressed as linear combinations of the columns of the matrix. In the case of an observable matrix, this means that there are certain states that cannot be described in terms of the variables used to construct the matrix. These states are indistinguishable from one another, because they cannot be distinguished by the observations available from the matrix.