Question

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

• A. If U is a 1-dimensional subspace of $\R^3$, then the null space of M is a 1-dimensional subspace.
• B. If U is a 2-dimensional subspace of $\R^3$, then the null space of M is a 1-dimensional subspace.
• C. M2 = M
• D. M3 = M

Answer 1

Answer: (B), (C), (D)

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

Answer 2

The correct option is (B) : If U is a 2-dimensional subspace of $\R^3$, then the null space of M is a 1-dimensional subspace, (C) : M2 = M and (D) : M3 = M.