List of top Linear Algebra Questions on Linear Algebra

Consider P: Let \( M \in \mathbb{R}^{m \times n} \) with \( m>n \geq 2 \). If \( \text{rank}(M) = n \), then the system of linear equations \( Mx = 0 \) has \( x = 0 \) as the only solution. Q: Let \( E \in \mathbb{R}^{n \times n}, n \geq 2 \) be a non-zero matrix such that \( E^3 = 0 \). Then \( I + E^2 \) is a singular matrix. Which of the following statements is TRUE?

Let \( M \) be a \( 3 \times 3 \) real matrix such that \( M^2 = 2M + 3I \). If the determinant of \( M \) is \( -9 \), then the trace of \( M \) equals __________.

Consider \( \mathbb{R}^3 \) as a vector space with the usual operations of vector addition and scalar multiplication. Let \( x \in \mathbb{R}^3 \) be denoted by \( x = (x_1, x_2, x_3) \). Define subspaces \( W_1 \) and \( W_2 \) by \[ W_1 := \{ x \in \mathbb{R}^3 : x_1 + 2x_2 - x_3 = 0 \} \] \[ W_2 := \{ x \in \mathbb{R}^3 : 2x_1 + 3x_3 = 0 \}. \] Let \( \text{dim}(U) \) denote the dimension of the subspace \( U \). Which of the following statements are TRUE?

Let A be a \(3 \times 4\) matrix and B be a \(4 \times 3\) matrix with real entries such that \( AB \) is non-singular. Consider the following statements:
P: Nullity of A is 0.
Q: \( BA \) is a non-singular matrix.
Then:

Let \( \langle \cdot, \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \) be an inner product on the vector space \( \mathbb{R}^n \) over \( \mathbb{R} \). Consider the following statements: P: \( |\langle u, v \rangle| \leq \frac{1}{2} \left( \langle u, u \rangle + \langle v, v \rangle \right) \) for all \( u, v \in \mathbb{R}^n \). Q: If \( \langle u, v \rangle = \langle 2u, -v \rangle \) for all \( v \in \mathbb{R}^n \), then \( u = 0 \). Then, which of the following is correct?