List of top Linear Algebra Questions

Consider the following statements :
I. Every infinite group has infinitely many subgroups.
II. There are only finitely many non-isomorphic groups of a given finite order.
Then

How many group homomorphisms are there from \(\Z_2\) to S₅ ?

From the additive group Q to which one of the following groups does there exist a non-trivial group homomorphism ?

Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A},   rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?

Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE ?

Let G be a finite group. Then G is necessarily a cyclic group if the order of G is

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

Which one of the following is TRUE for the symmetric group S₁₃ ?

Which of the following is a subspace of the real vector space \(\R^3\) ?

Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE ?

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ W.
Which one of the following holds ?

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6 ?

Consider the group G = {A ∈ M₂ (ℝ): AA^T = I₂} with respect to matrix multiplication. Let
Z(G) = {A ∈ G : AB = BA, for all B ∈ G}.
Then, the cardinality of Z(G) is

Let V be a nonzero subspace of the complex vector space 𝑀₇(ℂ) such that every nonzero matrix in 𝑉 is invertible. Then, the dimension of V over ℂ is

Which of the following is/are true ?

Which of the following is/are linear transformations ?

Let T : \(P_2(\R) → P_4(\R)\) be the linear transformation given by T(p(x)) = p(x²). Then the rank of T is equal to __________.

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

For g ∈ \(\Z\), let \(\bar{g}\) ∈ \(\Z_8\) denote the residue class of g modulo 8. Consider the group \(\Z^×_8\) = {\(\bar{x}\) ∈ \(\Z_8\) : 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from \(\Z^×_8\) onto itself is equal to ________

The number of permutations in S₄ that have exactly two cycles in their cycle decompositions is equal to ___________.

Let
\(A=\begin{pmatrix} 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 3 \\ 1 & 1 & 4 & 4 &4 \\ \end{pmatrix}\)
and B be a 5 × 5 real matrix such that AB is the zero matrix. Then the maximum possible rank of B is equal to ____________.

Let W be the subspace of \(M_3(\R)\) consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.

The maximum number of linearly independent eigenvectors of the matrix
\(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}\)
is equal to _________.

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?