Question:
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 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 __________.
Updated On: Jan 19, 2025
Hide Solution
Verified By Collegedunia
Correct Answer: 3
Solution and Explanation
The correct answer is 3.
Was this answer helpful?
0
1
Top Questions on Finite Dimensional Vector Spaces
- 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 ?- IIT JAM MA - 2024
- Linear Algebra
- Finite Dimensional Vector Spaces
- 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 ?- IIT JAM MA - 2024
- Linear Algebra
- Finite Dimensional Vector Spaces
- 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 _____________- IIT JAM MA - 2024
- Linear Algebra
- Finite Dimensional Vector Spaces
- 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
sr(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 a5,k ∈ \(\R\), then a5,2 equals ___________ (rounded off to two decimal places)- IIT JAM MA - 2024
- Linear Algebra
- Finite Dimensional Vector Spaces
- For k ∈ \(\N\), let 0 = t0 < t1 < ⋯ < tk < tk+1 = 1. A function f : [0,1] → ℝ is said to be piecewise linear with nodes t1, … ,tk, if for each j = 1, 2, … , k + 1, there exist aj ∈ ℝ, bj ∈ ℝ such that
f(t) = aj + bjt for tj-1 < t < tj.
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 ______________- IIT JAM MA - 2024
- Linear Algebra
- Finite Dimensional Vector Spaces
View More Questions
Questions Asked in IIT JAM MA exam
- Let y : ℝ → ℝ be the solution to the differential equation
\(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=1\)
satisfying y(0) = 0 and y'(0) = 1.
Then, \(\lim\limits_{x \rightarrow \infin}y(x)\) equals __________ (rounded off to two decimal places).- IIT JAM MA - 2024
- Differential Equations
- Consider the function f: ℝ → ℝ given by f(x) = x3 - 4x2 + 4x - 6.
For c ∈ ℝ, let
\(S(c)=\left\{x \in \R : f(x)=c\right\}\)
and |S(c)| denote the number of elements in S(c). Then, the value of
|S(-7)| + |S(-5)| + |S(3)|
equals _________- IIT JAM MA - 2024
- Functions of One Real Variable
- For a > b > 0, consider
\(D=\left\{(x,y,z) \in \R^3 :x^2+y^2+z^2 \le a^2\ \text{and } x^2+y^2 \ge b^2\right\}.\)
Then, the surface area of the boundary of the solid D is- IIT JAM MA - 2024
- Functions of Two or Three Real Variables
- For a positive integer \( n \), let \( U(n) = \{ r \in \mathbb{Z}_n : \gcd(r, n) = 1 \} \) be the group under multiplication modulo \( n \). Then, which one of the following statements is TRUE?
- IIT JAM MA - 2024
- Functions of One Real Variable
- 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?
- IIT JAM MA - 2024
- Functions of One Real Variable
View More Questions