List of top Linear Algebra Questions

Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation given by \( T(x, y) = (-x, y). \) Then
Let \( T : \mathbb{R}^7 \to \mathbb{R}^7 \) be a linear transformation with \( \text{Nullity}(T) = 2. \) Then, the minimum possible value for \( \text{Rank}(T^2) \) is ............

Let \( S^1 = \{z \in \mathbb{C} : |z| = 1\} \) be the circle group under multiplication and \( i = \sqrt{-1}. \) Then the set \( \{\theta \in \mathbb{R} : (e^{i2\pi\theta}) \text{ is infinite}\} \) is
 

Suppose that \( G \) is a group of order 57 which is not cyclic. If \( G \) contains a unique subgroup \( H \) of order 19, then for any \( g \notin H \), the order of \( g \) is ................
Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................
Let \( \phi : S_3 \to S_1 \) be a non-trivial non-injective group homomorphism. Then the number of elements in the kernel of \( \phi \) is .............
Consider the real vector space \( P_{2020} = \left\{ \sum_{i=0}^n a_i x^i : a_i \in \mathbb{R}, \, 0 \le n \le 2020 \right\}. \) Let \( W \) be the subspace given by \[ W = \left\{ \sum_{i=0}^n a_i x^i \in P_{2020} : a_i = 0 \text{ for all odd } i \right\}. \] Then the dimension of \( W \) is ................
Let \( V \) be a non-zero vector space over a field \( F \). Let \( S \subset V \) be a non-empty set. Consider the following properties of \( S \):
(I) For any vector space \( W \) over \( F \), any map \( f : S \to W \) extends to a linear map from \( V \) to \( W \).
(II) For any vector space \( W \) over \( F \) and any two linear maps \( f, g : V \to W \) satisfying \( f(s) = g(s) \) for all \( s \in S \), we have \( f(v) = g(v) \) for all \( v \in V \).
(III) \( S \) is linearly independent.
(IV) The span of \( S \) is \( V \).
Which of the following statement(s) is/are true?
No. of group homomorphism from Z4 to S4