Solution:

(a) Spanning Set for $V \subseteq \mathbb{R}^3$:

The vectors in $V$ are of the form: $v = (c_1, c_2, c_2 - 2c_1).$ Express $v$ in terms of $c_1$ and $c_2$: $v = c_1(1, 0, -2) + c_2(0, 1, 1).$ Thus, the vectors $(1, 0, -2)$ and $(0, 1, 1)$ span $V$. The spanning set for $V$ is: $\{ (1, 0, -2), (0, 1, 1) \}.$

(b) Spanning Set for $V \subseteq \mathbb{R}^4$:

The vectors in $V$ are of the form: $v = (c_1, c_2, c_2 - c_1, c_1 - 2c_2).$ Express $v$ in terms of $c_1$ and $c_2$: $v = c_1(1, 0, -1, 1) + c_2(0, 1, 1, -2).$ Thus, the vectors $(1, 0, -1, 1)$ and $(0, 1, 1, -2)$ span $V$. The spanning set for $V$ is: $\{ (1, 0, -1, 1), (0, 1, 1, -2) \}.$

(c) Spanning Set for the Subspace of $\mathbb{R}^3$:

The subspace consists of all solutions to the linear system: $x - 2y - z = 0.$ Rewriting, $x = 2y + z$. Let $y = t$ and $z = s$, where $t, s \in \mathbb{R}$. Then: $(x, y, z) = (2t + s, t, s).$ Expressing this in terms of $t$ and $s$: $(x, y, z) = t(2, 1, 0) + s(1, 0, 1).$ Thus, the vectors $(2, 1, 0)$ and $(1, 0, 1)$ span the subspace. The spanning set is: $\{ (2, 1, 0), (1, 0, 1) \}.$

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Summary of Spanning Sets:

1. (a) $\{ (1, 0, -2), (0, 1, 1) \}$.
2. (b) $\{ (1, 0, -1, 1), (0, 1, 1, -2) \}$.
3. (c) $\{ (2, 1, 0), (1, 0, 1) \}$.

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Would you like detailed verification or examples for any of the parts? Here are related questions to explore:

1. What is the definition of a spanning set, and why is it sufficient for linear combinations to generate the vector space?
2. How can we verify that the given sets are linearly independent?
3. Can we extend these spanning sets to bases for the respective vector spaces?
4. What is the geometric interpretation of these subspaces in $\mathbb{R}^3$ and $\mathbb{R}^4$?
5. How would the spanning set change if additional constraints were added to the systems?

Tip: To confirm a set spans a space, check if every vector in the space can be written as a linear combination of the set elements.