List of top Linear Algebra Questions

If \( \{ v_1, v_2, v_3 \} \) is a linearly independent set of vectors in a vector space over \( \mathbb{R} \), then which one of the following sets is also linearly independent?

Let \( m, n \in \mathbb{N}, m < n, P \in M_{n \times m}(\mathbb{R}), Q \in M_{m \times n}(\mathbb{R}) \). Which of the following is (are) NOT possible? 
 

Let \( W_1 \) be the real vector space of all \( 5 \times 2 \) matrices such that the sum of the entries in each row is zero. Let \( W_2 \) be the real vector space of all \( 5 \times 2 \) matrices such that the sum of the entries in each column is zero. Then the dimension of the space \( W_1 \cap W_2 \) is .................
Suppose \( Q \in M_{3 \times 3}(\mathbb{R}) \) is a matrix of rank 2. Let \( T : M_{3 \times 3}(\mathbb{R}) \to M_{3 \times 3}(\mathbb{R}) \) be the linear transformation defined by

\[ T(P) = QP. \]

Then the rank of \( T \) is ..................
No. of group homomorphism from Z4 to S4