To prove that $\dim(\text{span}(v_1 + w, v_2 + w, \dots, v_m + w)) \geq m - 1$, we will approach the problem by considering two cases based on the linear independence of the vectors $v_1, v_2, \dots, v_m \in V$.

Case 1: The vectors $v_1, v_2, \dots, v_m$ are linearly independent

In this case, since the vectors are linearly independent, their span has dimension $m$, i.e., $\dim(\text{span}(v_1, v_2, \dots, v_m)) = m$.

Now, let's examine the set $\{v_1 + w, v_2 + w, \dots, v_m + w\}$. Since $v_1, v_2, \dots, v_m$ are linearly independent, and adding $w$ to each vector does not affect their linear independence significantly. However, it is possible that the vectors $v_1 + w, v_2 + w, \dots, v_m + w$ are no longer linearly independent due to the presence of the common vector $w$.

The most that could happen is that the dimension of the span decreases by at most 1 because the vectors $v_1 + w, v_2 + w, \dots, v_m + w$ are now potentially linearly dependent (as they share a component involving $w$). Therefore, in this case: $\dim(\text{span}(v_1 + w, v_2 + w, \dots, v_m + w)) \geq m - 1.$

Case 2: The vectors $v_1, v_2, \dots, v_m$ are linearly dependent

If $v_1, v_2, \dots, v_m$ are linearly dependent, then $\dim(\text{span}(v_1, v_2, \dots, v_m)) < m$. Let's say that the dimension of the span is $k$, where $k < m$. So, there exist at least some linear dependencies between the vectors.

Now, consider the set $\{v_1 + w, v_2 + w, \dots, v_m + w\}$. The addition of $w$ to each vector can either:

1. Not affect the span significantly, in which case the dimension of $\text{span}(v_1 + w, v_2 + w, \dots, v_m + w)$ remains at least $k \geq m - 1$, or
2. Potentially increase the dimension by "breaking" some of the dependencies between the vectors by the presence of $w$.

In either case, we have: $\dim(\text{span}(v_1 + w, v_2 + w, \dots, v_m + w)) \geq m - 1.$

Conclusion:

In both cases (whether the original vectors $v_1, v_2, \dots, v_m$ are linearly independent or linearly dependent), we have shown that the dimension of the span of $v_1 + w, v_2 + w, \dots, v_m + w$ is at least $m - 1$. Therefore, the statement is proved: $\dim(\text{span}(v_1 + w, v_2 + w, \dots, v_m + w)) \geq m - 1.$

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Would you like further clarification or additional details on this proof? Here are some related questions that might interest you:

1. What is the difference between linearly dependent and independent vectors?
2. How does adding a common vector $w$ affect the span of a set of vectors?
3. Can you provide an example where the dimension decreases by exactly 1?
4. How is the span of a set of vectors related to their linear independence?
5. In what cases would the dimension increase when adding a vector $w$?

Tip: When analyzing the span of vectors, always consider how linear dependencies among the vectors influence the dimension.