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).
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).
Updated On: Jan 25, 2025
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Correct Answer: 3
Solution and Explanation
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:
\[
\text{dim}(V) = 3.
\]
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