For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers.  (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.)  Find the sum of all such alternating sums for $n = 7$.
1. **Understanding the Problem:**
   We need to find the sum of all alternating sums for the set $\{1, 2, 3, \dots, 7\}$. The alternating sum for a subset is defined by arranging the numbers in decreasing order and then alternately adding and subtracting them.

2. **Analyzing the Alternating Sum:**
   Let's denote the set $\{1, 2, 3, \dots, 7\}$ as $S$. For any subset $T \subseteq S$, we define the alternating sum as follows:
   - Arrange the elements of $T$ in decreasing order.
   - Alternately add and subtract the elements starting with the largest.

3. **Considering Subsets Containing 7:**
   Let $T$ be a subset of $S$ that contains the element 7. Define $T' = T \setminus \{7\}$, which is the subset $T$ without the element 7. Notice that $T'$ is also a subset of $S$.

4. **Relationship Between Alternating Sums of $T$ and $T'$:**
   The alternating sum of $T$ can be written as:
   \[
   \text{AltSum}(T) = 7 - a_1 + a_2 - a_3 + \cdots \pm a_k
   \]
   where $a_1, a_2, \ldots, a_k$ are the elements of $T'$ arranged in decreasing order.

   The alternating sum of $T'$ is:
   \[
   \text{AltSum}(T') = a_1 - a_2 + a_3 - \cdots \pm a_k
   \]

   Adding these two sums, we get:
   \[
   \text{AltSum}(T) + \text{AltSum}(T') = (7 - a_1 + a_2 - a_3 + \cdots \pm a_k) + (a_1 - a_2 + a_3 - \cdots \pm a_k) = 7
   \]

5. **Counting Subsets Containing 7:**
   There are $2^6$ subsets of $S$ that contain the element 7, because each of the remaining 6 elements can either be included or not included in the subset independently.

6. **Summing the Alternating Sums:**
   Since each pair $(T, T')$ contributes 7 to the total sum, and there are $2^6$ such pairs, the total sum of all alternating sums for subsets containing 7 is:
   \[
   7 \times 2^6 = 7 \times 64 = 448
   \]

The final answer is $\boxed{448}$