Question

Let \( W_1 \) and \( W_2 \) be subspaces of the real vector space \( \mathbb{R}^{100} \) defined by \[ W_1 = \{ (x_1, x_2, \dots, x_{100}) : x_i = 0 \text{ if } i \text{ is divisible by } 4 \}, \] \[ W_2 = \{ (x_1, x_2, \dots, x_{100}) : x_i = 0 \text{ if } i \text{ is divisible by } 5 \}. \] Then the dimension of \( W_1 \cap W_2 \) is

Hint:

When finding the dimension of the intersection of subspaces defined by coordinate conditions, count the number of independent constraints for each subspace and subtract from the total dimension.

Answer 1

Correct Answer: 60

Step 1: Analyze the conditions for \( W_1 \) and \( W_2 \).
In \( W_1 \), the coordinates corresponding to multiples of 4 are zero. Since there are 25 multiples of 4 in \( \{1, 2, \dots, 100\} \), the dimension of \( W_1 \) is \( 100 - 25 = 75 \). In \( W_2 \), the coordinates corresponding to multiples of 5 are zero. Since there are 20 multiples of 5 in \( \{1, 2, \dots, 100\} \), the dimension of \( W_2 \) is \( 100 - 20 = 80 \).
Step 2: Find the intersection of \( W_1 \) and \( W_2 \).
The intersection \( W_1 \cap W_2 \) contains the coordinates where both conditions are true: the coordinates corresponding to multiples of both 4 and 5 (i.e., multiples of 20) must be zero. There are 5 such coordinates (because \( 100 / 20 = 5 \)). Thus, the dimension of the intersection is \( 100 - 5 - 15 = 25 \), since 5 coordinates are constrained by both conditions, and 15 additional coordinates are constrained by either condition.
Step 3: Conclusion.
Thus, the dimension of \( W_1 \cap W_2 \) is \( \boxed{25} \).