Let V be a complex vector space such that dim V = n and let T E L(V) be diagonalizable. We define the subspace U C L(V) by, U = {S € L(V) : ST = TS} Prove that dim U 2 n.

Q: Let f(x) = 5, g(r) = 8x – 5 and h(r) = 82?. Consider the inner product (P, 9) = / P(z)9(z) dz in the…

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Q: Let S = { (x, y, z, w) E R*; x – 2y + w = 0 and 2x + z – w = 0 } . - Prove that S is a subspace by…

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Q: (b) If V = R", and U is a subspace of V of dimension m with 0 <m <n, prove that U² = {v € V: B(u, v)…

A: As per our guidelines we are supposed to solve only one problem. For the solution of the first…

Q: Let H be the set of points inside and on the unit circle in the xy-plane. That is, let {{]***a :x² +…

A: According to the given information, H be the set of points inside and on the unit circle in the x-y…

Q: Let W1 and W2 be subspaces of a vector space V having dimensions m and n, respectively, where m > n.…

A: Given, dimension of W1=mdimension of W2=n such that m≥n   We know, W1∩W2 is a subspace of W1W1∩W2 is…

Q: 3) Suppose TE L(V) and U1, . , Um are subspaces of V invariant under T. Prove that U, ...+ Um is…

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Q: b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U…

A: In a finite-dimensional Vector space, every subspace is finite-dimensional.

Q: Let W be the subspace of R4 spanned by the vectors 1 1 and -6 -1 6 1 4 Find the matrix A of the…

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Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

Q: f(x) = 9, g(x) = 7x – 4 and h(x) = -2x². Consider the inner product (p, q) = / p(z)q(x) dæ in the…

A: Let S={v1,v2,v3} represent the orthonormal basis for the subspace. By Gram Schmidt Process,…

Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

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Q: If V is the vector space of real valued continuous functions , then show that the set W of all d’y…

A: Our Aim is to show that the set W of all solutions of the differential equation3d2ydx2-5dydx+7y=0…

Q: Let V = {(x, y, 2x – 3y)|x, yɛR}. Prove that V is closed for addition and scalar multiplication i.e.…

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Q: Consider the vector space V = {p(x) = ao +ajr+azx² | a¡ € R, deg(p(æ)) < 2} of polynomials of degree…

A: Let          V(x)=p(x)=a0+a1x+a2x2 | ai∈ℝ, deg(p(x))≤2 of polynomials of degree less than or equal…

Q: Let W = {a + bx + cx + dx| a+ b = 0 and c – 3d = 0} be a subspace of P3 Then the dimension of W is…

A: Linear combination of vectors we use to find.

Q: 3. For a certain subspace W, the orthogonal projection of the vector y = |4 onto W 5 is ŷ = | 0 Use…

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Q: Let H be the set of all points in the second and fourth quadrants in the plane V=R^2. That is, H={…

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Q: Let W be a subspace of Rn. Prove that the intersection of W and W⊥ is {0}, where W⊥ is the subspace…

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Q: et X = {(x,y, z): Vx² + y² + z² <1} be a subspace R° with the standard topology and 0 = (0,0,0).…

A: Given,               X=x, y, z:x2+y2+z2 ≤1  be a subspace of R3 with the standard topology and o=0,…

Q: Let W = {a + bx + cx? + dx³ € Pg : c = a – b+ 2d} be a subspace of P3. Then a basis for W is:

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Q: Let V = P2(t) with inner product defined by (f, g) = So f(t)g(t) dt. Find a basis for the subspace W…

A: An inner product is a generalization of the dot product. A real vector space, an inner product…

Q: Let S be the subspace of Rn spanned by the vectors x1, x2, . . . , xk. Show that y ∈ S⊥ if and only…

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Q: For a certain subspace W, the orthogonal projection of the vector y = 3 onto W is ŷ 2. Use this…

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Q: In the vector space of functions f: R → R, find a basis for the subspace spanned by {sin t, sin 2t ,…

A: Solving question by theorem vector space

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.…

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Q: Let V be a real vector space, EC V a subspace, and T : V → a linear transformation satisfying T(E) C…

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Q: Suppose T E L(V) and U is a subspace of V. (a) Prove that if U null T, then U is invariant under T.…

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Q: Let W = :a +5c = 0} be a subspace of M22. %3D Then dimension of W is equal to: None of the mentioned…

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Q: Let W = (a + bx+ cx + dx'|a+ b= 0 and c -3d = 0 ) be a subspace of P Then the dimension of W is…

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Q: Let u, = 2 |, u2 and uz = 0 Note that u, and uz are orthogonal but that uz is not orthogonal to u,…

A: The orthogonal vector is given as follows :

Q: 4) Let X, Y, Z be subspaces of a vector space V and assume that Y CX. Prove that Xn(Y+Z)% =Y+(XnZ).…

A: 4. Given The modular law of sub-space: Let X,Y,Z be subspaces of a vector space V and assume that…

Q: Let W = {a + bx + cx2 + dx'|c-3d = 0} be a subspace of P, Then the dimension of W is equal to %3D 4.…

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Q: Find an orthonormal basis for the subspace of C(R) generated by the functions {1, x, x²} with…

A: W e find an orthonormal basis for the subspace of C([0, 1]) generated by the function (1, x, x^2)

Q: Let W fa + bx + cx2 + dx'| a+ b 0 and c-3d 0 } be a subspace of Pg. %3D %3D %3D Then the dimension…

A: a+b=0  and c-3d=0 b=-a and c=3d Equation given is; a+ bx+ cx2+ dx3  = a- ax+ 3dx2+ dx3 that can be…

Q: Consider the following linear operator; T : R - R given by T(x. y, z) (-y, x +2, z). Find a…

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Q: Let W = (a + bx + cx2 + dx'| c- 3d = 0} be a subspace of P. Then the dimension of W is equal to O…

A: dim(W) = number of vector in the basis of W.

Q: Suppose V is finite-dimensional, with dim V = n > 1. Prove that there exist 1-dimensional subspaces…

A: Let us consider that (v1, v2, ..., vn) be a basis for V, and let Ui= span(vi), for i ∈ {1, 2, ...,…

Q: Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V,…

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Q: 2. Find a basis for the subspace W = {[r y z]T: ax + by + cz = 0, a, b, c e R} of R". What is the…

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Q: Suppose that V is a vector space with basis B= {bi | i E I} and S is a subspace of V. Let {B1,... ,…

A: Given that V is a vector space with basis B=bi|i∈I and S is a subspace of V. Let B1,B2,B3,…,Bk be a…

Q: For a fixed a ϵ R, determine the dimension of the subspace of Pn(R) defined by {f ϵ Pn(R): f(a) =…

A: Assume a ϵ R and the set V=f∈PnR: fa=0 is subspace. Now, assume the arbitrary polynomial f from V.…

Q: Let F be the vector space of infinitely differentiable functions from R to R. Consider the subspace:…

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Q: Use the inner product (f, 9) = ". f(x)g(x) dx in the vector space Co[0, 1] of continuous functions…

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Q: Let V be a subspace of M22 with basis 2 0 0] By 3 and let W be a vector space with basis Bw (х, у,…

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Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: For a certain subspace W, the orthogonal projection of the vector y = 5 onto W 4 is ŷ = |1. Use this…

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Q: Prove that S = {(x, y, z) E R³: 2x + 3y+z = 0} is a subspace of the vector space R3.

A: Explanation of the answer is as follows

Q: Let V - P2 be the vector space of polynomials of degree at most 2, and let H be the span of the…

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Q: Let V=R^3 and let H be the subset of V of all points on the plane -3x+5y-5z=0. Is H a subspace of…

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