Question

Consider \( \mathbb{R}^3 \) as a vector space with the usual operations of vector addition and scalar multiplication. Let \( x \in \mathbb{R}^3 \) be denoted by \( x = (x_1, x_2, x_3) \). Define subspaces \( W_1 \) and \( W_2 \) by \[ W_1 := \{ x \in \mathbb{R}^3 : x_1 + 2x_2 - x_3 = 0 \} \] \[ W_2 := \{ x \in \mathbb{R}^3 : 2x_1 + 3x_3 = 0 \}. \] Let \( \text{dim}(U) \) denote the dimension of the subspace \( U \). Which of the following statements are TRUE?

• A. \( \text{dim}(W_1) = \text{dim}(W_2) \)
• B. \( \text{dim}(W_1) + \text{dim}(W_2) - \text{dim}(\mathbb{R}^3) = 1 \)
• C. \( \text{dim}(W_1 + W_2) = 2 \)
• D. \( \text{dim}(W_1 \cap W_2) = 1 \)

Hint:

For subspaces of vector spaces, use the formula for the dimension of the sum of two subspaces: \[ \text{dim}(W_1 + W_2) = \text{dim}(W_1) + \text{dim}(W_2) - \text{dim}(W_1 \cap W_2). \]

Answer 1

The Correct Option is A, B, D

We are given two subspaces \( W_1 \) and \( W_2 \) of \( \mathbb{R}^3 \), and we need to check the validity of the provided statements. Step 1: Finding the dimension of \( W_1 \) and \( W_2 \)
The equation defining \( W_1 \) is \( x_1 + 2x_2 - x_3 = 0 \). This is a linear equation in three variables, so the solution space has dimension 2. Hence, \( \text{dim}(W_1) = 2 \). Similarly, the equation defining \( W_2 \) is \( 2x_1 + 3x_3 = 0 \). This is also a linear equation in three variables, so the solution space has dimension 2. Thus, \( \text{dim}(W_2) = 2 \). Step 2: Checking statement (A)
Since \( \text{dim}(W_1) = 2 \) and \( \text{dim}(W_2) = 2 \), statement (A) is true: \[ \text{dim}(W_1) = \text{dim}(W_2). \] Step 3: Checking statement (B)
Using the formula for the dimension of the sum of two subspaces: \[ \text{dim}(W_1 + W_2) = \text{dim}(W_1) + \text{dim}(W_2) - \text{dim}(W_1 \cap W_2). \] We can now compute the dimension of the intersection \( W_1 \cap W_2 \). Step 4: Finding the intersection of \( W_1 \) and \( W_2 \)
To find \( \text{dim}(W_1 \cap W_2) \), we solve the system of equations: \[ x_1 + 2x_2 - x_3 = 0 \] \[ 2x_1 + 3x_3 = 0. \] Solving these equations, we find that the intersection \( W_1 \cap W_2 \) is a one-dimensional subspace. Therefore, \( \text{dim}(W_1 \cap W_2) = 1 \). Now we can compute \( \text{dim}(W_1 + W_2) \): \[ \text{dim}(W_1 + W_2) = 2 + 2 - 1 = 3. \] Thus, the equation \( \text{dim}(W_1) + \text{dim}(W_2) - \text{dim}(\mathbb{R}^3) = 1 \) is valid, since \( \text{dim}(\mathbb{R}^3) = 3 \). Therefore, statement (B) is true. Step 5: Checking statement (C)
We have already computed \( \text{dim}(W_1 + W_2) = 3 \), so statement (C) is false because \( \text{dim}(W_1 + W_2) \neq 2 \). Step 6: Checking statement (D)
We computed \( \text{dim}(W_1 \cap W_2) = 1 \), so statement (D) is true. Step 7: Conclusion
The correct answers are: - (A) \( \text{dim}(W_1) = \text{dim}(W_2) \), - (B) \( \text{dim}(W_1) + \text{dim}(W_2) - \text{dim}(\mathbb{R}^3) = 1 \), - (D) \( \text{dim}(W_1 \cap W_2) = 1 \). Thus, the correct answer is (A), (B), (D).