LINEAR ALGEBRA PLEASE ANSWER ALL QUESTIONS Consider the vector space V = C with scalar multiplication over the real numbers R, and le T:V → V be the linear operator defined by T (z,, z2) = (z, – iz2, Zz – Z2). Let W be the cyclic subspace of V generated by w = (1 + 2i, 1 + 5. 1 Find the T-cyclic basis 3 for W generated by w. 2 Find the characteristic polynomial of Tw. 3 Find (Tw]a. 4 Explain whether T = Tw.

Q: Let (:/ be an inner product in the vector space V. Given an isomorphism1' : U + [u, v] = (Tu, Tv),…

A: Let , be an inner product in the vector space V and  given an isomorphism T : U→V.…

Q: Let V = R². For (u₁, U2), (v1, v2₂) € V and a € R define vector addition by (U1, U2) = (v1, v2) :=…

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Q: Let (:) be an inner product in the vector space V. Given an isomorphismT :U → V. Score (u, v) = (Tu,…

A: Given , is an inner product on vector space V over real numbers. Therefore for all u,v,w∈V and a∈ℝ:…

Q: Explain why S is not a basis for R². s={(6, 8), (1, 0), (0, 1)) S is linearly dependent. OS does not…

A: We will use S to determine which among the given options is correct.

Q: Let V be a vector space over R. Suppose {v, w, ż} is a basis for V. Prove that {+w, - w, ż} is also…

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Q: Let V = R². For (u₁, U2), (v₁, v₂) € V and a ER define vector addition by (u₁, U₂) (V1₁, V₂) = = (U₁…

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Q: Let {u1, let B be a basis for V. Let S vectors of {u1,·· , Un} with respect to basis B. Prove that ,…

A: Given V is a n- dimensional vector space. Let B be a basis for V. Let B=v1,v2,…,vn. Let u1, u2, …,…

Q: Consider the line V = {(t,t + 1) |t € R} in R. Let V be the real vector space with vector addition e…

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Q: Consider the line V = {(t,t + 1)|t e R} in R. Let V be the real vector space with vector addition e…

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Q: Let P2 be the vector space of polynomials of degree 2 or less. Consider the following two ordered…

A: In mathematics, an ordered basis of a vector space of finite dimension n allows representing…

Q: Check if set X is linearly i ndependent, spans r3 and check for basis of r3 X={(1,0,0),(0,1,0),…

A: A set is a BASIS if it linearly independent and spans the vector space

Q: Which of the follwing vectors span R2 Let V be the Vector space R? 1.1], [-1,1]

A: Remark: A set of vectors S={s1, s2, s3,..., sn} is linearly independent if and only the equation…

Q: (c) If S = {V₁, V₂, ..., Vn} is a set of vectors in a finite-dimensional vector space V, then S is…

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Q: If V is the space of all vectors in R*that are orthogonal to W = Span{(1,0,2,1), (0,0,1,5)}, then…

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Q: Let V be the vector space of functions spanned by B = {bị = 6 sin æ + 4 cos a, b2 = 2 sin æ + 7 cos…

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Q: Next question If B is the standard basis of the space P3 of polynomials, then let B= {1, t, t“, t°}.…

A: A matrix coordinate vectors are linearly independent if there is a pivot in each column.

Q: Determine whether the vector {6,1,–6, 2} is in the vector space generated by the basis vectors {1,1,…

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Q: Let B = (v₁.....V₁) be a linearly independent set in R". Explain why B must be a basis for R".

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Q: Let V be a vector space of dimension 4, and let W be a vector space of dimension 2. Let L:VW be a…

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Q: Let (, ) be an inner product in the vector space V. Given an isomorphismT : U H V, Put [u, v] = (Tu,…

A: Given , is an inner product on vector space V over real numbers. Therefore for all u,v,w∈Vand a∈ℝ:…

Q: If vi, v2, ..., Vn are independent vectors in R", then they must form a basis of R". True False…

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Q: 3. Let P2 be the vector space of polynomials of degree at most 2. Define a map D: P2 → P2, it sends…

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Q: 2 Determine whether the vectors form a basis for R*. (You may use any relevant theorem that we have…

A: A basis for a subspace S of Rn is a set of vectors in S that is linearly independent and is maximal…

Q: Let W be the set of all vectors with x and y real. Find a basis of W-

A: Here, W is a set of vectors of the form xyx+y=x101+y011 with x and y are real numbers. Now, the…

Q: Let V be a vector space and u and w be vectors in V. Prove that {u, w} is linearly independent if…

A: Linearly independent sets: A set of vectors {v1,v2.......vn} is said to be linearly independent if…

Q: A: Prove that a vector (2,-1, 1) is in the vector space W =span((2, 1, 3), (4, 0, 4)) and that (3,…

A: Vector space, W=span2, 1, 3, 4, 0, 4Vectors : 2, -1, 1 and 3, 2, 7

Q: --- V₁ = 6 y V₂ Then B is a basis of: 12, B={₁,₂). H = Gen (v1, v2} Because v1 and v2 are linearly…

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Q: The two bases B1 (which needs not be R) are related in the following manner. {V1, V2, V3} and B2 =…

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Q: -H 3 { and C { Let B } be two bases of R2 Find the change of coordinate matrix P_ sending a B…

A: Required to determine the change of basis matrix from B to C.

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…

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Q: Let B, = {v,, v,,v, } be a basis for a vector space V. Determine whether or not B, = {ū,,ū,,ūz} is a…

A: Let V be vector space over field F . S⊂V(F) is said to be basis if S is Linearly independent Linear…

Q: The set of vectors below is drawn from the Euclidean space R; S = {(-1,1,0), (2, 2, 1), (0, 1, 1),…

A: Vectors of S are: (-1,1,0),(2,2,1),(0,1,1),(2,3,-2) Since it is a set of 4 vectors in R3, it cannot…