Problem Analysis

We are given four vectors $v_1, v_2, v_3, v_4 \in \mathbb{R}^4$ such that:

$v_i = \begin{pmatrix} v_i^{(1)} \\ v_i^{(2)} \\ v_i^{(3)} \\ v_i^{(4)} \end{pmatrix} \quad \text{for } i = 1, 2, 3, 4,$ and the condition:

$\sum_{i=1}^4 v_i^{(j)} = 0 \quad \text{for each } j = 1, 2, 3, 4.$

We are tasked with determining the possible dimensions of the subspace $W$ of $\mathbb{R}^4$ spanned by $\{v_1, v_2, v_3, v_4\}$, and identifying which dimension $W$ cannot have.

Step 1: Understanding the condition

The condition:

$\sum_{i=1}^4 v_i^{(j)} = 0 \quad \text{for each } j = 1, 2, 3, 4$

means that the sum of the $i$-th components of all the vectors is zero. This implies a linear dependence among the vectors. Specifically, the four vectors $v_1, v_2, v_3, v_4$ cannot all be linearly independent in $\mathbb{R}^4$.

Step 2: Maximum possible dimension of $W$

• $\{v_1, v_2, v_3, v_4\}$ is a set of at most 4 vectors in $\mathbb{R}^4$, so the dimension of $W$ can be at most 4.
• However, the condition $\sum_{i=1}^4 v_i^{(j)} = 0$ imposes constraints, reducing the linear independence of the vectors.

Step 3: Minimum possible dimension of $W$

Since the vectors are not trivial (not all zero), the minimum possible dimension of $W$ is 1.

Step 4: Using the dependency condition

The condition $\sum_{i=1}^4 v_i^{(j)} = 0$ creates a dependency among the vectors. Specifically, this condition can be rewritten as:

$v_4 = -(v_1 + v_2 + v_3).$

Thus, $v_4$ is determined by $v_1, v_2,$ and $v_3$. Therefore, $\{v_1, v_2, v_3\}$ can span all of $W$, making the dimension of $W$ at most 3.

Step 5: Conclusion

The dimension of $W$ cannot be 4 due to the dependency among the vectors. Therefore, the answer is:

$\boxed{\text{(c) 3}}$

Follow-up Questions

1. Why does the condition $\sum_{i=1}^4 v_i^{(j)} = 0$ imply linear dependence among the vectors?
2. Can the dimension of $W$ be 1, and under what conditions?
3. How would the solution change if the vectors were in $\mathbb{R}^5$ instead of $\mathbb{R}^4$?
4. What is the general relationship between the number of vectors and their span in $\mathbb{R}^n$?
5. Can you construct explicit examples of such vectors satisfying the given condition?

Tip

To determine the dimension of a subspace spanned by a set of vectors, always check for linear dependencies and apply the rank-nullity theorem or dependency relationships.