Question

Let \( V = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1 = x_2 \} \). Consider \( V \) as a subspace of \(\mathbb{R}^4\) over the real field. Then the dimension of \( V \) is equal to __________ (answer in integer).

Answer 1

Correct Answer: 3

1. Constraint on \( V \): - The condition \( x_1 = x_2 \) imposes a single linear constraint on the elements of \( \mathbb{R}^4 \). - This reduces the number of free variables from 4 to 3. 2. Basis for \( V \): - A basis for \( V \) can be constructed by choosing three independent vectors that satisfy \( x_1 = x_2 \), such as: \[ \{(1, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)\}. \] 3. Dimension of \( V \): - The number of basis vectors is 3, so: \[ \text{dim}(V) = 3. \]