LetV = R[x]3, W = R[x]6be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree ≤ 6 respectively.This question concerns the map α : V → W which sends a polynomial f(x) in V to f(x^2) in W.(a) Show that α is linear as a map V →W.(b) Find the nullity ν(α) and the rank ρ(α).(c) Let U = Image(α) ⊆ W and find a subspace Z ⊆ W such that W = U ⊕ Z. LetV = R[x] 3, W = R[x]6be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree< 6 respectively.This question concerns the map a : V → W which sends a polynomial f(x) in V tof(x²) in W.(a) Show that a is linear as a map V → W.(b) Find the nullity v(a) and the rank p(a).(c) Let U = Image(a) CW and find a subspace ZC W such that W = UZ.Justify your answers.

Answered: Image /qna-images/answer/ee1eac55-9145-48c0-b572-14a9c8ea3161.jpg

V = R[x] 3, W = R[x]6 be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree 6 respectively. This question concerns the map :: V→ W which sends a polynomial f(x) in V to f(x²) in W. (a) Show that a is linear as a map V → W. (b) Find the nullity v(a) and the rank p(a). (c) Let U = Image(a) C W and find a subspace ZC W such that W = U Z.

V = R[x] 3, W = R[x]6 be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree 6 respectively. This question concerns the map :: V→ W which sends a polynomial f(x) in V to f(x²) in W. (a) Show that a is linear as a map V → W. (b) Find the nullity v(a) and the rank p(a). (c) Let U = Image(a) C W and find a subspace ZC W such that W = U Z.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: (4) Suppose V and W are finite dimensional vector spaces such that dim(V) > dim(W). Show that any…

Q: Find the dimension of the vector space U of all linear transformations of V into W for each of the…

A: a)- Given: V=R2 , W=R3 Since R2 is 2-dimensional vector space and R3 is 3-dimensional vector space,…

Q: Let V be the vector space of all functions from R into R; let V\index{e} be the subset of even…

A: Let V be the vector space of all functions from R into R. and Ve = { f(x) : f(-x)…

Q: Suppose V1, ..., Vn are vector spaces then does L(V1 x...x Vn, U) have the same dim with L(V1, U)…

Q: Let T: V-->W be a linear transformation between finite-dimensional vector spaces V and W Let B…

A: It is given that, V and W be the finite-dimensional vector spaces. And, T : V→W be a linear…

Q: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

Q: Let P, denote the vector space of polynomials in the variable of degree n or less with real…

Q: (i) Consider the vector space M2×2 with the trace inner product ( A,B) = trace(A'B) and let A = .…

A: We are entitled to solve only one question at a time. Given - To find -

Q: Let T: V-->W be a linear transformation between finite-dimensional vector spaces V and W Let B…

A: Let V and W be vector spaces and B and C are bases for V and W respectively and let T be linear…

Q: Let P3(R) denote the R-vector space of real polynomials of degree 2 or less p=a+bX+cX2, where a, b,…

A: Given the polynomial .Let the subspace of be .The subspace is the set of all polynomials from such…

Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

Q: Let T : V → V be a linear operator over a ﬁnite dimensional vector space V . Prove that dim(ker(T))…

Q: Let P, denote the vector space of polynomials in the variable of degree or less with real…

Q: Let F be the vector space of all functions mapping R into R. Determine whether or not the given…

Q: 3) Let V be a vector space and denote Vk: := V × ... × V as the cartesian product of itself k times.…

Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)…

A: Given be the set of all pairs of real numbers together with the following operations:

Q: Let V be the vector space of functions spanned by Nπ B = {b₁ = 5 sin x + 7 cos x, b₂ = 3 sin x + 9…

Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

Q: 3. Let V be the vector space of polynomials with coefficients from R. a) Show that the following…

Q: Let u and v be nonzero orthogonal vectors in R3. Find bases for the range space and null space of…

A: To find: To find the bases for the range space and null space using the given function f: ℝ3→ℝ3,…

Q: If V(F) is a vector space of polynomials in t of degree at most 3 and D be the differentiation…

A: Given D is differentiation transformation. The basis is 1, t, t2, 1+t3. T(1)=ddt1=0…

Q: Let V be the vector space of all polynomials in the variable x over R. Show that the mappings…

Q: Show that {t, e'}, which is a subset of F, the vector space containing all real-valued functions, is…

Q: For all differentiable vector fields v(r) and w(r), use index notation to show that - . - . v × (▼ ×…

A: We have to show .

Q: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

Q: Let S = {v1, ... , vn} be a linearly independent set in a vector space V. Show that if v is a vector…

Q: Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients,…

Q: Let V be the vector space of functions spanned by B = {b₁ = 6 sin x + 7 cos x, b₂ = 5 sin x + 3 cos…

Q: : Let V be the set of all ordered pairs of real numbers (u1, u). Consider the following addition and…

Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V…

Q: Let P, is the vector space of polynomials of the degrees2.Use coordinate vectors to that the…

A: The main objective is to check {p1,p2,p3} linear independent and basis of P2.

Q: Let C[0, 1] be the vector space of continuous functions that are defined on the interval [0, 1].…

Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

Q: Let x1(t), x2(t), . . . , xn (t) be vector functions whose ith components (for some fixed i )…

A: Given that x1(t), x2(t), . . ., xn(t) be vector functions whose ith components ( for some fixed i)…

Q: Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

A: Let V be a vector space and suppose there exist maps S,T:V→V such that .We need to prove that the…

Q: Show how a real

A: To Prove that how a real linear functional u on a complex linear normed space gives rise to a…

Q: Find a linear mapping G that maps [0, 1] × [0, 1] to the parallelogram in the xy-plane spanned by…

A: The linear map G maps to the parallelogram in XY-plane.The parallelogram is spanned by the vectors…

Q: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Question

100%

Let
V = R[x]3, W = R[x]6

be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree ≤ 6 respectively.

This question concerns the map α : V → W which sends a polynomial f(x) in V to f(x^2) in W.

(a) Show that α is linear as a map V →W.

(b) Find the nullity ν(α) and the rank ρ(α).

Let
V = R[x] 3, W = R[x]6
be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree
< 6 respectively.
This question concerns the map a : V → W which sends a polynomial f(x) in V to
f(x²) in W.
(a) Show that a is linear as a map V → W.
(b) Find the nullity v(a) and the rank p(a).
(c) Let U = Image(a) CW and find a subspace ZC W such that W = UZ.
Justify your answers.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: This Theorem will apply

VIEW

Step 2: Let us begin the solution

VIEW

Step 3: finding rank and nullity

VIEW

Step 4: answer of (c)

VIEW

Solution

VIEW

Step by step

Solved in 5 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Does the set of all functions that vanish at x = 0 and x = L form a vector space? If it does, explicitly show that it satisfies all eight properties required of a vector space. If not, which property fails? Show how it fails.
Find a linear mapping G that maps [0, 1] x [0, 1] to the parallelogram in the xy-plane spanned by the vectors (-3,7) and (9,3). (Use symbolic notation and fractions where needed. Give your answer in the form (*, *).) G(u, v) =
Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map f: EF, the following properties are = n. equivalent: (a) f is bijective. (b) f is surjective. (c) f is injective.. (d) Ker f = (0).
Let V be a vector space and T: V->V be a linear map such that T^2=T. 1) Prove that lm(T)=ker(Id-T) 2) Prove that V=ker(T)⊕Im(T)
Let V be the vector space of functions spanned by B = {b₁ = 2 sin x + 9 cos x, b₂ = 4 sin x + 7 cos x} where x C = {C₁ = sin x, C₂ = cos } where x of coordinates matrix P. C+B P = C←B [f(x)] c = [ The coordinates of a function f(x) relative to the basis B are [f(x)] B = [8] 6 coordinates of f(x) relative to C and find f(x). f(x) = Ex: 3 Ex: 3 nn 2 sin x + -, ne Z nn 2,n € Z is also a basis for V. Find the change COS x Find the
7. Let V be the vector space of polynomials in two variables x and y of degree at most two: V = {ax² + bxy + cy² + dx + ey + ƒ | a, b, c, d, e, ƒ € R}. Let T be the linear operator on V defined by T(g(x, y)) = Ə Ix 9 (x, y) + Find the Jordan canonical form of T. Ə dy I (x, y). ду
5. Let P2 be the vector space of polynomials of degree at most 2. Determine if the vector v = 10x – 7 is in the subspace Span{3x? + x – 1, x² – 3x + 1}
4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).
In the vector space P (polynomial space) consider the linear operators D, A : P+ P defined by D(p(x))=p'(x) A(p(x))=xp(x) Find the sets Ker(D) and Im(D), Ker(A) and Im(A)
2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].