Suppose v1 through vm is linearly independent in V and w is in V. Prove that dim(span(v1+w,....,vm+w)) is greater than or equal to m-1.

Steps to Solve

1. Consider the span of the original vectors

Let $S = {v_1, v_2, \dots, v_m}$ be a linearly independent set in the vector space $V$. Thus, $\text{dim}(\text{span}(S)) = m$. We are given the set $S' = {v_1 + w, v_2 + w, \dots, v_m + w}$, where $w \in V$, and we want to show that $\text{dim}(\text{span}(S')) \ge m-1$.

2. Examine the case when w is in the span of S

If $w \in \text{span}(S)$, then $w = \sum_{i=1}^m a_i v_i$ for some scalars $a_i$. In this case, each $v_i + w$ is still a linear combination of $v_1, \dots, v_m$, so $\text{span}(S') \subseteq \text{span}(S)$. However, we cannot definitively say that the dimension of the span is greater than or equal to $m-1$ in this particular instance without further analysis of dependencies introduced.

3. Consider differences between the vectors in S'

Consider the vectors $v_2 + w - (v_1 + w) = v_2 - v_1$, $v_3 + w - (v_1 + w) = v_3 - v_1$, ..., $v_m + w - (v_1 + w) = v_m - v_1$. These $m-1$ vectors, $v_2 - v_1, v_3 - v_1, \dots, v_m - v_1$ are in the span of $S'$.

4. Prove linear independence of the difference vectors

We want to show that the $m-1$ vectors $v_2 - v_1, v_3 - v_1, \dots, v_m - v_1$ are linearly independent. Suppose that $$c_2(v_2 - v_1) + c_3(v_3 - v_1) + \dots + c_m(v_m - v_1) = 0$$ Rearranging the terms, we have $$- (c_2 + c_3 + \dots + c_m)v_1 + c_2 v_2 + c_3 v_3 + \dots + c_m v_m = 0$$ Since ${v_1, v_2, \dots, v_m}$ are linearly independent, we must have $$-(c_2 + c_3 + \dots + c_m) = 0, \quad c_2 = 0, \quad c_3 = 0, \quad \dots, \quad c_m = 0$$ From $c_2 = c_3 = \dots = c_m = 0$, the first equation is also satisfied. Therefore, the vectors $v_2 - v_1, v_3 - v_1, \dots, v_m - v_1$ are linearly independent.

5. Relate linear independence to dimension

Since we have found $m-1$ linearly independent vectors in $\text{span}(S')$, the dimension of $\text{span}(S')$ must be at least $m-1$. Therefore, $\text{dim}(\text{span}(S')) \ge m-1$.