List of top Linear Algebra Questions

Let P₁₁(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 11, together with the zero polynomial. Let
E = {s₀(x), s₁(𝑥), … , s₁₁(x)}, F = {r0 (x), r₁ (x), … , r₁₁(x)}
be subsets of P₁₁(x) having 12 elements each and satisfying
s₀ (3) = s₁ (3) = ⋯ = s₁₁(3) = 0 , r₀(4) = r₁(4) = ⋯ = r₁₁(4) = 1.
Then, which one of the following is TRUE ?

Let A be a 6 × 5 matrix with entries in ℝ and B be a 5 × 4 matrix with entries in ℝ. Consider the following two statements.
P : For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 × 4 zero matrix.
Q : For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 × 5 zero matrix.
Which one of the following holds ?

Let P₇(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 7, together with the zero polynomial.Let T : P₇(x) → P₇(𝑥) be the linear transformation defined by
\(T(f(x))=f(x)+\frac{df(x)}{dx}\).
Then, which one of the following is TRUE ?

Let \(a=\begin{bmatrix} \frac{1}{\sqrt3} \\ \frac{-1}{\sqrt2} \\ \frac{1}{\sqrt6} \\ 0\end{bmatrix}\). Consider the following two statements.
P : The matrix I₄ - aa^T is invertible.
Q: The matrix I₄ - 2aa^T is invertible.
Then, which one of the following holds ?

Which of the following statements is/are TRUE ?

The center Z(G) of a group G is defined as
z(G) = {x ∈ G ∶ xg = gx for all g ∈ G}.
Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G ?

For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, C ∈ M10(\(\R\)) satisfying A = BC ?

Consider
\(𝐺 = \left\{𝑚 + 𝑛\sqrt2\ ∶\ 𝑚, 𝑛 ∈ \Z\right\}\)
as a subgroup of the additive group ℝ.
Which of the following statements is/are TRUE ?

Consider the 4 × 4 matrix
\(M = \begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{pmatrix}\)
If a_i,j denotes the (i, j)^th entry of M^-1 , then a_4,1 equals __________ (rounded off to two decimal places).

Let P₁₂(x) be the real vector space of polynomials in the variable x with real coefficients and having degree at most 12, together with the zero polynomial. Define
\(𝑉 = \left\{𝑓 ∈ 𝑃_{12}(𝑥): 𝑓(−𝑥) = 𝑓(𝑥) \text{for all}\ 𝑥 ∈ ℝ\ \text{and}\ 𝑓(2024) = 0\right\}.\)
Then, the dimension of V is _____________

Let P₇(x) be the real vector space of polynomials in x with degree at most 7, together with the zero polynomial. For r = 1, 2, … , 7, define
s_r(x) = x(x - 1) ⋯ (x − (r - 1)) and s₀(x) = 1.
Consider the fact that B = {s₀(x), s₁(x), … , s₇(x)} is a basis of P₇(𝑥).
If
\(x^5=\sum\limits_{k=0}^7a_{5,k}s_k(x),\)
where a_5,k ∈ \(\R\), then a_5,2 equals ___________ (rounded off to two decimal places)

Let
\(M=\begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 2 & 0 & 0 & 0 & -4 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 3 \\ 0 & 0 & 0 & 2 & 2 \end{pmatrix}\)
If p(x) is the characteristic polynomial of M, then p(2) - 1 equals _________

For k ∈ \(\N\), let 0 = t₀ < t₁ < ⋯ < t_k < t_k+1 = 1. A function f : [0,1] → ℝ is said to be piecewise linear with nodes t₁, … ,t_k, if for each j = 1, 2, … , k + 1, there exist a_j ∈ ℝ, b_j ∈ ℝ such that
f(t) = a_j + b_jt for t_j-1 < t < t_j.
Let V be the real vector space of all real valued continuous piecewise linear functions on [0,1] with nodes \(\frac{1}{4},\frac{1}{2},\frac{3}{4}\) . Then, the dimension of V equals ______________

Consider the following statements :
I. There exists a linear transformation from \(\R^3\) to itself such that its range space and null space are the same.
II. There exists a linear transformation from \(\R^2\) to itself such that its range space and null space are the same.
Then

Let
\(\begin{pmatrix} 1 & -1 & 0 \\ 0 & 0 & 0 \\ -2 & 2 & 2 \end{pmatrix}\)
and B = A⁵ + A⁴ + I₃. Which of the following is NOT an eigenvalue of B ?

The system of linear equations in x₁, x₂, x₃
\(\begin{pmatrix} 1 & 1 & 1 \\ 0 & -1 & 1 \\ 2 & 3 & \alpha \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} 3 \\ 1 \\ \beta \end{pmatrix}\)
where α, β ∈ R, has

Let S and T be non-empty subsets of \(\R^2\), and W be a non-zero proper subspace of \(\R^2\). Consider the following statements :
I. If span(S) = \(\R^2\) , then span(S ∩ W) = W.
II. span(S ∪ T) = span(S) ∪ span(T).
Then

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