Given that $S_1$ and $S_2$ are subspaces and $S_1 \subseteq S_2$, along with the fact that the dimension of $S_2$, $\dim(S_2) = 4$, we can analyze the possible dimensions of $S_1$.

1. What are the possible dimensions of $S_1$?

Since $S_1 \subseteq S_2$ and $S_1$ is nonzero, the dimension of $S_1$, denoted $\dim(S_1)$, must be a nonzero integer that is less than or equal to the dimension of $S_2$. Hence, the possible dimensions of $S_1$ are:

$\dim(S_1) = 1, 2, 3, \text{ or } 4$

These are the possible values for $\dim(S_1)$, as $S_1$ is a subspace of $S_2$ and cannot have a dimension greater than $S_2$'s dimension.

2. If $S_1 \neq S_2$, then what are the possible dimensions of $S_1$?

If $S_1 \neq S_2$, the dimension of $S_1$ must be strictly less than the dimension of $S_2$, which means $\dim(S_1) < \dim(S_2) = 4$. The possible dimensions of $S_1$ in this case are:

$\dim(S_1) = 1, 2, \text{ or } 3$

These are the values $\dim(S_1)$ can take if $S_1$ is strictly contained in $S_2$ and not equal to it.

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Would you like further explanation on any of these answers?

Related Questions:

1. What is the definition of a subspace in linear algebra?
2. How does the dimension theorem (rank-nullity theorem) relate to subspaces?
3. Can two subspaces of the same dimension be equal if one is contained in the other?
4. How does the intersection of two subspaces affect their dimensions?
5. What are examples of real-life applications where subspaces and their dimensions are important?

Tip:

Remember that a subspace of a vector space must contain the zero vector and be closed under addition and scalar multiplication.