Let X={a,b,c} and B={{a,c},{a},{b}}. Let T be the topology generated by B and consider A={a,b}. With respect to T, compute the closure A¯¯¯ of A and the boundary ∂A=A¯¯∩(X∖A)¯¯ of A. Let T be the topology generated by B and consider B={a,c} viewed as a subspace of X. Is there a set that is closed relative to B that is not closed in X? Justify your answer.

Q: Let Y be a vector subspace of a normed space X. Show that the closure of Y is also a vector…

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

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Q: Let X = {x, y, z} and B = {{x, z}, {x}, {y}}. Let T be the topology generated by B and consider B =…

A: Given: T is topology generated by B=x, z, x, y and B=x, z is a subspace of a topological space (X,…

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Given W be a subspace spanned by the u's, We have to write y as the sum of a vector in W and a…

Q: Suppose that U and V are both subspaces of R . Prove that their intersection U ∩ V = {v : v ∈ U…

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Q: Let (X, F) be a space. Prove that a subspace of a subspace is a subspace of (X, F).

A: Here given X, I is a topological space.

Q: Let W be a subspace of a vector space V. Show that if x is in both W and W+, then x = 0.

A: Solution is given below

Q: a. The union of two subspaces of a vector space is always a subspace. b. The basis for the zero…

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Q: Let V be a vector space with subspaces U and W. Prove multiplication that       U ⋂W is a subspace…

A: Given:V is a vector space with subspaces U and W. To Prove:U∩W is a subspace of V.

Q: Determine whether the set W={(s, t, s + t) : s and t are real number} is a subspace of vector space…

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Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

Q: c) Mark each statement True or False. Justify the answer. A subspace is also a vector space. • R? is…

A: Here , A subspace is olso a vector space: it is true.  R^2 is a subspace of R^3 : it is false.

Q: V={(x,y,z1):x=y}, V = {(x,y,) *x+y= 2,1=2y} için bir baz bulunuz.

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Q: Let V be the two-dimensional subspace of R4 spanned by (0, 1, 0, 1) and (0, 2, 0, 0). Write the…

A: To find v, we need to find the projection of u onto the subspace V, which can be found using the…

Q: Determine If The Subset W Is A Vector Subspace Of V = R3 w = {(x, y, z): x * y-z = 0

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

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Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

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Q: a) ir(A - 1)1- ( -2) %3D Find the matrix A

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Q: Consider the set W={(x, y, x+y) | x, y are real}. Is Wa subspace of R³? If so, prove it. Otherwise…

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Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

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Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that ,  spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

Q: Let (R, Tus) be the real line.Let r ∈ R such that 0 ≤ r < 1 and consider I = [0, 1] as a subspace of…

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Q: Let V be a finite dimensional vector space and W CV a subspace. Prove that W has a complimentary…

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Q: (R, Ts) is a topological space and NC R , find the T. subspace topology.(T|N is the subspace…

A: Let ℝ, τs is a topological space and ℕ⊆ℝ Define τs|ℕ=ℕ∩A: A∈τs Then τs|ℕ is a topology on ℕ  (i) As…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.