Questions: Let W be the subset of R^3 consisting of all vectors [ x1 x2 x3 ] such that x1+x2+x3>2 Determine if W is a subspace of R^3 and check the correct answer(s) below. A. W is a subspace because it can be written as N(A) for some matrix A. B. W is a subspace because it can be expressed as W=spanv1, ..., vn. C. W is not a subspace because it is not closed under scalar multiplication. D. W is not a subspace because it does not contain the zero vector.

Solution Steps

To determine if \( W \) is a subspace of \( \mathbb{R}^3 \), we need to check if it satisfies the following conditions:

1. It contains the zero vector.
2. It is closed under vector addition.
3. It is closed under scalar multiplication.

Given the condition \( x_1 + x_2 + x_3 > 2 \), we can immediately see that the zero vector \(\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]\) does not satisfy this inequality. Therefore, \( W \) does not contain the zero vector, and hence, it cannot be a subspace of \( \mathbb{R}^3 \).

• Check if the zero vector is in \( W \).
• Conclude that \( W \) is not a subspace if the zero vector is not in \( W \).

To determine if \( W \) is a subspace of \( \mathbb{R}^3 \), we first need to check if the zero vector \(\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]\) is in \( W \). The condition for a vector to be in \( W \) is \( x_1 + x_2 + x_3 > 2 \).

For the zero vector: \[ 0 + 0 + 0 = 0 \] Since \( 0 \) is not greater than \( 2 \), the zero vector is not in \( W \).

Since the zero vector is not in \( W \), \( W \) cannot be a subspace of \( \mathbb{R}^3 \). One of the necessary conditions for a subset to be a subspace is that it must contain the zero vector.

Final Answer