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Problem 1

# The discrete-time system $$\Sigma$$ is linear (over the field $$\mathbb{K}$$ ) if: \- It is complete; \- $$X$$ and $$\mathcal{U}$$ are vector spaces; and \- $$\mathcal{P}(t, \cdot \cdot \cdot)$$ is linear for each $$t \in \mathbb{Z}$$. The system with outputs $$\Sigma$$ is linear if in addition: \- $$y$$ is a vector space; and \- $$h(t, \cdot)$$ is linear for each $$t \in \mathbb{Z}$$. The system is finite dimensional if both $$\mathcal{U}$$ and $$X$$, as well as $$y$$ for a system with outputs, are finite dimensional; the dimension of $$\Sigma$$ is in that case the dimension of $$\mathcal{X}$$.

Expert verified
A discrete-time system Σ is linear if it is complete, and the input, state, and output spaces ($$\mathcal{U}$$, $$\mathcal{X}$$, and y) are vector spaces. Additionally, the operators $$\mathcal{P}(t, \cdot, \cdot)$$ and $$h(t, \cdot)$$ must be linear for each $$t \in \mathbb{Z}$$. The system is finite-dimensional if the input, state, and output spaces have finite dimensions, with the dimension of Σ being the same as the dimension of the state space $$\mathcal{X}$$.
See the step by step solution

## Step 1: Define a discrete-time system

A discrete-time system is a mathematically described system operating over discrete time instants. It is represented by Σ and takes input from a set of signals or systems U, has a state space X, and gives output in y.

## Step 2: Define what it means for a system to be complete

A discrete-time system is complete if it has a unique solution for every initial condition in X and input sequence in U.

## Step 3: Linear systems and their requirements

The given exercise defines a discrete-time system to be linear if it satisfies the following conditions: 1. The system is complete. 2. The input and state spaces, denoted by $$\mathcal{U}$$ and $$\mathcal{X}$$, are vector spaces. 3. The operator $$\mathcal{P}(t, \cdot, \cdot)$$ representing the system's dynamics is linear for each $$t \in \mathbb{Z}$$ (for every discrete-time instant).

## Step 4: Explain linear systems with outputs

A discrete-time system with outputs is considered linear if, in addition to the conditions mentioned in step 3, the output space y is a vector space, and the operator $$h(t, \cdot)$$ representing the output mapping is linear for each $$t \in \mathbb{Z}$$ (for every discrete-time instant).

## Step 5: Define finite-dimensional systems and their dimensions

A discrete-time system is considered finite-dimensional if both the input and state spaces, as well as the output space for a system with outputs, have finite dimensions. In such a case, the dimension of the system, denoted by Σ, corresponds to the dimension of the state space $$\mathcal{X}$$.

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