Let's analyze each statement one by one to determine which ones are true.

(A) If S is linearly independent, any set containing S is also linearly independent.

• False. If $S$ is linearly independent, adding more vectors to $S$ can potentially make the set linearly dependent. For example, if $S = \{ \mathbf{v}_1, \mathbf{v}_2 \}$ is linearly independent, adding a vector $\mathbf{v}_3 = \mathbf{v}_1 + \mathbf{v}_2$ makes the set $\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}$ linearly dependent.

(B) If S spans V, any set containing S also spans V.

• True. If $S$ spans $V$, any set that contains $S$ will span $V$ because any vector in $V$ can be written as a linear combination of the vectors in $S$, and adding more vectors to the set will still allow for a linear combination that spans the entire space $V$.

(C) If S is linearly dependent, any set containing S is also linearly dependent.

• True. If $S$ is linearly dependent, this means there exists a nontrivial linear combination of the vectors in $S$ that equals the zero vector. Adding more vectors to this set does not eliminate this dependence, so the larger set will also be linearly dependent.

(D) If S spans V, any subset of S also spans V.

• False. If $S$ spans $V$, it is not necessarily true that any subset of $S$ will also span $V$. A minimal spanning set (also known as a basis) spans $V$, but a subset of $S$ could miss important vectors that are necessary to span the entire space.

Correct statements:

• (B) If $S$ spans $V$, any set containing $S$ also spans $V$.
• (C) If $S$ is linearly dependent, any set containing $S$ is also linearly dependent.

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Would you like a more detailed explanation on any of these points, or have any other questions?

Related questions:

1. What is the difference between a linearly independent and a linearly dependent set of vectors?
2. How do we prove that a set spans a vector space?
3. What is a basis of a vector space, and how does it relate to spanning sets?
4. How do you find the minimal set of vectors that spans a space (a basis)?
5. What happens if you add redundant vectors to a linearly independent set?

Tip: A basis is a minimal spanning set, meaning it's a set of vectors that spans the space but is also linearly independent.