This paper proves that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This result is applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The paper also discusses the importance of determining when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts, which is central to subspace system identification. The paper introduces the concept of persistency of excitation in the context of behavioral systems. It defines a linear time-invariant system as a triple (T, W, B), where T is the time axis, W is the signal space, and B is the behavior. The behavior is defined as a linear, shift-invariant, and complete set of signals. The paper shows that a behavior B ∈ L^w can be represented by a kernel representation, which is a matrix of polynomials in the indeterminate ξ. The module of annihilators of B is defined as the set of polynomials that annihilate B. The paper discusses several integer invariants associated with L^w, including the variable cardinality, input cardinality, output cardinality, state cardinality, lag, and shortest lag. These invariants are computed from a kernel representation and are important for understanding the properties of the system. The main result of the paper is Theorem 1, which states that if the input component of the observed signal is persistently exciting of order L + n, where n is the state cardinality, then the left kernel of the Hankel matrix of the observed signal equals the annihilator module of the system, and the row span of the Hankel matrix equals the behavior of the system over the window length L. The paper also provides corollaries and special cases, including the case of a state space system. It discusses the conditions under which the system matrices can be recovered from input/state/output trajectories and the number of data points needed to identify the system. The paper concludes with acknowledgments of the financial support received for this research.This paper proves that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This result is applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The paper also discusses the importance of determining when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts, which is central to subspace system identification. The paper introduces the concept of persistency of excitation in the context of behavioral systems. It defines a linear time-invariant system as a triple (T, W, B), where T is the time axis, W is the signal space, and B is the behavior. The behavior is defined as a linear, shift-invariant, and complete set of signals. The paper shows that a behavior B ∈ L^w can be represented by a kernel representation, which is a matrix of polynomials in the indeterminate ξ. The module of annihilators of B is defined as the set of polynomials that annihilate B. The paper discusses several integer invariants associated with L^w, including the variable cardinality, input cardinality, output cardinality, state cardinality, lag, and shortest lag. These invariants are computed from a kernel representation and are important for understanding the properties of the system. The main result of the paper is Theorem 1, which states that if the input component of the observed signal is persistently exciting of order L + n, where n is the state cardinality, then the left kernel of the Hankel matrix of the observed signal equals the annihilator module of the system, and the row span of the Hankel matrix equals the behavior of the system over the window length L. The paper also provides corollaries and special cases, including the case of a state space system. It discusses the conditions under which the system matrices can be recovered from input/state/output trajectories and the number of data points needed to identify the system. The paper concludes with acknowledgments of the financial support received for this research.