GATE CS 2007 Q27 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To determine which option is true, let us analyze the set $X$ and the given set of vectors.

1. Verify if $X$ is a subspace

The set $X$ is defined by the homogeneous linear equation:

This can be represented as the null space of the $1 \times 3$ matrix $A = [1, 1, 1]$. Since the null space of any matrix is a subspace of its domain, $X$ is a subspace of $\mathbb{R}^3$. This makes Option C false.

2. Find the dimension of $X$

Using the Rank-Nullity Theorem for the matrix $A$:

Thus, any basis of $X$ must consist of exactly $2$ linearly independent vectors belonging to $X$.

3. Check the given set of vectors

Let $v_1 = [1, -1, 0]^T$ and $v_2 = [1, 0, -1]^T$.

• Membership in $X$:
  • For $v_1$: $1 + (-1) + 0 = 0 \implies v_1 \in X$
  • For $v_2$: $1 + 0 + (-1) = 0 \implies v_2 \in X$
• Linear Independence:

  Since neither vector is a scalar multiple of the other, the set $\{v_1, v_2\}$ is linearly independent.

Conclusion

Since $\{v_1, v_2\}$ is a set of $2$ linearly independent vectors in a $2$-dimensional subspace $X$, it automatically spans $X$ and forms a basis for $X$.

Therefore, Option A is TRUE.