Problem-5

vector spaces
subspaces

Question

Let \(M_{2 \times 2}(\mathbb{R})\) represent the set of all \(2 \times 2\) real matrices. Now consider the following sets: \[ \begin{align*} V_1 &= \{A\ |\ A \in M_{2 \times 2}(\mathbb{R}) \text{ and } A \text{ is a diagonal matrix}\}\\ V_2 &= \{A\ |\ A \in M_{2 \times 2}(\mathbb{R}) \text{ and sum of diagonal elements of } A \text{ is equal to 1}\} \end{align*} \]

  1. \(V_1\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\), but \(V_2\) is not

  2. \(V_2\) is a subspace of \(M_{2 \times 2}(\mathbb{R})\), but \(V_1\) is not

  3. Both \(V_1\) and \(V_2\) are subspaces of \(M_{2 \times 2}(\mathbb{R})\)

  4. Neither \(V_1\) nor \(V_2\) are subspaces of \(M_{2 \times 2}(\mathbb{R})\)

Solution

\(V_2\) does not have the zero element. \(V_1\) has the zero element, is closed under addition and scalar multiplication. The correct answer is (a).