The question is about the properties of a set of vectors $\{v_1, v_2, v_3, v_4\}$ forming a spanning set for a vector space $V$. We are asked to determine which statement is true about the set $\{v_1, v_2, v_3\}$.

Given the statement that $\{v_1, v_2, v_3, v_4\}$ forms a spanning set for the vector space $V$, this means that any vector in $V$ can be written as a linear combination of these four vectors.

Let’s analyze the options:

• Option (a): $\{v_1, v_2, v_3\}$ are linearly dependent in $V$.

  • Since $\{v_1, v_2, v_3, v_4\}$ is a spanning set and it contains four vectors, there might be redundancy among the vectors. It’s possible that $v_4$ can be written as a linear combination of $v_1, v_2,$ and $v_3$, which implies that $\{v_1, v_2, v_3\}$ could be linearly dependent.
  • This could be true.

• Option (b): $\{v_1, v_2, v_3\}$ are linearly independent in $V$.

  • If $\{v_1, v_2, v_3, v_4\}$ is a spanning set and if we can express $v_4$ as a linear combination of $v_1, v_2, v_3$, this would imply that $v_4$ adds no new information, and thus $\{v_1, v_2, v_3\}$ would be linearly dependent, not independent.
  • This is not true.

• Option (c): $\{v_1, v_2, v_3\}$ is a spanning set of $V$.

  • If $v_4$ can be written as a linear combination of $v_1, v_2, v_3$, this means that $v_1, v_2, v_3$ alone do not span $V$. Therefore, $\{v_1, v_2, v_3\}$ would not be a spanning set by itself.
  • This is not true.

• Option (d): $\{v_1, v_2\}$ is not a spanning set of $V$.

  • If $\{v_1, v_2, v_3, v_4\}$ spans $V$, then $\{v_1, v_2\}$ is clearly not sufficient to span the entire space, as the set would lack the vectors $v_3$ and $v_4$.
  • This is true.

Conclusion:

The correct answer is (a): $\{v_1, v_2, v_3\}$ are linearly dependent in $V$.

Would you like further clarification or have any follow-up questions? Here are a few related questions you might be interested in:

1. What is the definition of a linearly independent set in a vector space?
2. How can we determine if a set of vectors is linearly dependent or independent?
3. What is the role of a spanning set in the context of vector spaces?
4. How can you reduce a set of vectors to find a minimal spanning set?
5. How does the number of vectors in a spanning set relate to the dimension of a vector space?

Tip: When checking for linear dependence or independence, one effective method is to set up a system of equations to check if the only solution to a linear combination being zero is the trivial one (all coefficients are zero).