Question-49

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

• If \(U\) is a 1-dimensional subspace of \(\mathbb{R}^3\), then the null space of \(\mathbf{M}\) is a one-dimensional subspace.
• If \(U\) is a 2-dimensional subspace of \(\mathbb{R}^3\), then the null space of \(\mathbf{M}\) is a one-dimensional subspace.
• \(\mathbf{M}^2 = \mathbf{M}\)
• \(\mathbf{M}^3 = \mathbf{M}\)

Answer

• If \(U\) is a 1-dimensional subspace of \(\mathbb{R}^3\), then the null space of \(\mathbf{M}\) is a one-dimensional subspace.
• If \(U\) is a 2-dimensional subspace of \(\mathbb{R}^3\), then the null space of \(\mathbf{M}\) is a one-dimensional subspace.
• \(\mathbf{M}^2 = \mathbf{M}\)
• \(\mathbf{M}^3 = \mathbf{M}\)

Solution

• Since \(\mathbf{M}\) is a projection matrix, its column space is \(U\) and its null space is \(U^{\perp}\). If the dimension of \(U\) is \(k\), the dimension of the null space is \(3 - k\).
• For a projection matrix, \(\mathbf{M}^{2} = \mathbf{M}\). It follows that \(\mathbf{M}^{3} = \mathbf{M}\).