In each case, determine whether V is a vector space. If it is not a vector space, explain why not. If it is, find basis vectors for V. (a) V is the subset of R³ defined by 4x - 5y + z = 1. (b) Let the vector w = (w₁, W₂,.. , wn) represent a portfolio's holdings, where each component w, represents the fraction of the portfolio's total market value in as- set i. Let V be the set of weight vectors that can represent market-neutral long/short portfolios. The weights w; satisfy 0 < w; ≤ 1 for long positions, -1 ≤w, <0 for short positions, and Σω = 0. (c) V is the set of vectors in R2 for which Mv = v, where (2 3¹). M =

Q: If V is a vector space then show that cV=V.

A: Vector space: A non-empty set V closed with two binary operations called addition and scalar…

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A: The given matrix is A=1236. Obtain the reduced row echelon form of matrix A as follows…

Q: (a) A: va+v. (b) A: (av). (c) A: √ av.

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: Let v1 and v2 be vectors in an inner product space V.(a) Is it possible for |{v1, v2} | to be…

A: It is given that,  v1, v2 be the vectors in an inner product space V. a) We have to determine that…

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Q: Determine whether the vectors (3,0, 3, 6), (6,0, 0, 3), (0,0,3,0),(6,3,9,12) form a basis of R*.

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Q: State whether the following statement is true or false: A vector space may have two or more zero…

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Q: Suppose that {x1, x2, x3} is a linearly independent set of vectors in a vector space V . Must {x1,…

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Q: Show whether or not the set of vectors B = {U, V, W} where U = V = and W = 0 is a basis for R³.

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Q: (a) Give examples of three distinct vectors v1, v2, v3 in the vector space R3 such that v2 E…

A: Let v1 = {1,1,1}v2 = {2,3,4}v3 = {1,2,3} Here v2 = {2,3,4} = {1,1,1}+{1,2,3} = v1+v3

Q: Write the vector w = (1, 1, 1) as a linear combination of vectors in the set S. V2 V3 S = {(1, 2,…

A: See the attachment.

Q: Find the Euclidean norm for vector v = [2 3 1 -1]

A: We have to find the euclidean norm

Q: Find all values of a such that the vectors = and are perpendicular

A: Given:The vectors are  and  are perpendicular.To Find:The value of .Concept Used:When two vectors…

Q: or the given sets of vectors, find the dimension of the space spanned by them and basis for this…

A: In this question, we will find the dimension and basis of space.

Q: Show that W is a vector space.

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Q: Find a basis for the space spanned by these vectors:

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Q: Find the angles between each vector in A where the set of vectors A = { [1, 1, 1], [1, 1, 0], [1, 0,…

A: Question explained below :Explanation:ANSWER :​​

Q: The two vectors u=(-2,3,1,4) and v= (1,2,0, -1) are orthogonal in the vector space R4 True False

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Q: Are the vector spaces R4, M2,2, and M1,4 exactly the same? Describe their similarities and…

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Q: Let S be the set of all vectors in R¹ perpendicular to a vector space. Then find a basis of S. and…

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Q: Determine the dimension of the vector space given by -a V = | a,b E R a b

A: Given V = -aa0, -b0b | a, b ∈ℝ

Q: Explain an example of vector space.

A: In this question, we have to explain an example of vector space. Vector space :  An Abelian group…

Q: Let (V, (, )) be an inner product space and let v, w be vectors in V. If (v, w) = 7 and w = 5, then…

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Q: Consider the vector space 2 with the inner product rgk)=x+1,then %3D

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Q: Determine whether the given vectors (1, 1,-1, 1), (1,-1, 2,-1), (3, 1, 0, 1) in R* are lincarly…

A: Given vectors are: (1, 1, -1, 1) , (1, -1, 2, -1) , (3, 1, 0, 1) We have to determine whether the…

Q: Let V be a vector space and suppose that {u, v, w} is an independent set of vectors in V. For each…

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Q: Let W be the set of all vectors of the form where a and b are real numbers. 2a+b (a) Determine…

A: Given, W=ab2a+b where a and b are real numbers.

Q: Find the length of the projection of 3   onto the vector -3…

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Q: Determine whether the set of vectors {(2,-1,3),(4,1,2),(8,-1,8)} form a basis for R3.

A: Consider the vectors : 2,-1,3, 4,1,2, 8,-1,8 To form a basis these vectors, the vectors must be…

Q: In a vector space of dimension 5, every set of 6 vectors would span the vector space.

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Q: (i) Let {u, v} be linearly independent vectors in a vector space V. Determine whether set {v – u, v…

A: 7 (i) Given, u, v be linearly independent vectors in a vector space.

Q: Are all functions from one vector space to another linear?

A: Let f: V→W is a  map from vector  V to W such that fx=x2 The given example the function from one…

Q: Describe the zero vector (the additive identity) of the vector space. P3 x + ]+² +

A: We have to describe the zero vector (the additive identity ) of the vector space P3.