List of top Linear Algebra Questions on Finite Dimensional Vector Spaces asked in IIT JAM MA

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 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 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)

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 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

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

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 ?

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 __________.

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

Let \( M \) be a \( 4 \times 3 \) real matrix and let \( \{e_1, e_2, e_3\} \) be the standard basis of \( \mathbb{R}^3. \) Which of the following is true?

Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation given by \( T(x, y) = (-x, y). \) Then

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?

Let \( U, V \) and \( W \) be finite dimensional real vector spaces, \( T: U \to V \), \( S: V \to W \) and \( P: W \to U \) be linear transformations. If range \( (ST) = \) nullspace \( (P) \), nullspace \( (ST) = \) range \( (P) \) and rank \( (T) = \) rank \( (S) \), then which one of the following is TRUE?

Consider the vector space \( V \) over \( \mathbb{R} \) of polynomial functions of degree less than or equal to 3 defined on \( \mathbb{R} \). Let \( T : V \to V \) be defined by \( (T f)(x) = f(x) - x f'(x) \). Then the rank of \( T \) is

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 \( W_1 \) 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 ..................