GATE DA 2024 | Question 49

MSQ · 2 marks

Let \(\mathbb{R}\) be the set of real numbers, \(U\) be a subspace of \(\mathbb{R}^{3}\) and \(\boldsymbol{M} \in \mathbb{R}^{3 \times 3}\) be the matrix corresponding to the projection on to the subspace \(U\).

Which of the following statements is/are TRUE?

  1. Option A: If \(U\) is a 1-dimensional subspace of \(\mathbb{R}^{3}\), then the null space of \(\boldsymbol{M}\) is a 1-dimensional subspace.
  2. Option B: If \(U\) is a 2-dimensional subspace of \(\mathbb{R}^{3}\), then the null space of \(\boldsymbol{M}\) is a 1-dimensional subspace. (correct)
  3. Option C: \(\boldsymbol{M}^{2} = \boldsymbol{M}\) (correct)
  4. Option D: \(\boldsymbol{M}^{3} = \boldsymbol{M}\) (correct)

Answer

(B), (C), (D)

Explanation

A projection onto \(U\) leaves every vector of \(U\) unchanged and sends the complementary directions to \(\boldsymbol{0}\). So its column space is \(U\), and by rank–nullity its null space has dimension \(3 - \dim U\).

  • A fails: a line \(U\) leaves a null space of dimension \(3 - 1 = 2\), a plane.
  • B holds: a plane \(U\) leaves a null space of dimension \(3 - 2 = 1\), a line.
  • C holds: \(\boldsymbol{M}\boldsymbol{x}\) already lies in \(U\), so projecting it again changes nothing, which gives \(\boldsymbol{M}^{2} = \boldsymbol{M}\).
  • D holds: \(\boldsymbol{M}^{3} = \boldsymbol{M}^{2}\boldsymbol{M} = \boldsymbol{M}\boldsymbol{M} = \boldsymbol{M}\).

Answer: B, C, D