The GATE Grind

GATE 2024 DA – Question 49

Linear Algebra · Special matrices: projection, orthogonal and idempotent, quadratic forms · 2 marks · Multiple select

Let $\mathbb{R}$ be the set of real numbers, $U$ be a subspace of $\mathbb{R}^3$ and $M \in \mathbb{R}^{3 \times 3}$ be the matrix corresponding to the projection on to the subspace $U$. Which of the following statements is/are TRUE?

  1. If $U$ is a 1-dimensional subspace of $\mathbb{R}^3$, then the null space of $M$ is a 1-dimensional subspace.
  2. If $U$ is a 2-dimensional subspace of $\mathbb{R}^3$, then the null space of $M$ is a 1-dimensional subspace.
  3. $M^2 = M$
  4. $M^3 = M$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) If $U$ is a 2-dimensional subspace of $\mathbb{R}^3$, then the null space of $M$ is a 1-dimensional subspace.; (C) $M^2 = M$; (D) $M^3 = M$

Explanation

The rank of the projection matrix equals $\dim U$, so the null space has dimension $3 - \dim U$. For a 1-dimensional $U$ it is 2-dimensional, so A is false; for a 2-dimensional $U$ it is 1-dimensional, so B is true. A projection is idempotent, $M^2 = M$, so C is true, and then $M^3 = M^2M = M \cdot M = M$, so D is true.