Question 3. Let (X,r) be a topological space and YCX. Show that if the subspace (Y,r,) is connected, then the subspace (Y.r;) is connected.

Q: Show that the set W : {(x, y, 5x-4y)| x and y are real numbers} is a subspace of R³.

A: We are give, W=x,y,5x-4y| x and y are real numbers To show: W is a subspace of R3 This can be done…

Q: QUESTION 3 Let V be a vector space and let u, ve V. IfW= (au+Bela, ß e R), prove that W is a…

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Q: Find the dimension of the subspace W of R* spanned by set S = {(3,0,1,–2), (–5,4,9,2), (–1,2,5,0)} .

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Q: 2. Determine whether the set W = {(x,y, z)|x + y +z = 0} is a subspace of R3

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Q: CLEICE If (Y, Ty) is a subspace of (X, T). Then each open set in Y is open in X if I only if Y is…

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Q: Question 4. If U is a subspace of R", show that U-1 = U

A: we have U is a subspace of ℝn to show - U⊥⊥=U we know  U⊥=x∈ℝn: x·u=0 for every u∈UU⊥⊥=x∈ℝn: x·w=0…

Q: Question 4. If U is a subspace of R", show that U-- = U.

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Q: Suppose W,, W2,. (a) span (W, W2 (b) If S; spans W; for i = 1,...,r, then S, US, U...US, spans W, +…

A: (a) W1+W2W1+W2 would be W2W2. Below is the justification.  x+y∈W1+W2,wx+y∈W1+W2where x∈W1x∈W1 and…

Q: If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it

A: NOTE:  If V is a finite dimensional vector space and W is a subspace of V, then W is finite…

Q: , dim W 5. Let U, W be subspaces of a vector space V. If dim U and Un W = {0}, what is dim (U+ W)?

A: Let V be a vector space over a field k and let U, W be finite-dimensional subspaces of V.  Then…

Q: Question 2. Suppose U and W are subspaces of a vector space V with basis B and C, respectively. Show…

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Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: 14. Let V = R°and W = {(a, b, c) E Vla² + b² = c²}. Is W a subspace of V? If so,what is its…

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Q: If W is a subspace of R" of dimension k with a basis {v1,e,2,..., v%} then W Range(|v1 v2 vn)). ...…

A: Given: Let W be a subspace of Rn and dimk=k Let B be a basis of k then B=k Also B is a linearly…

Q: The set {(x, , x,)' 1 x,+ x, =0} form a subspace of R Select one: O True False

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Q: Let W { (а, b, а + 2b + с, 2а — Зc), а, ь, с€ R} %3D - 1. Show that W is a subspace of R4. 2.…

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Q: Is the subset V {(x, y, z, w), y = 3z, w = 2x + 1} a subspace of R*? Show the details of %3D your…

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Q: If S and T are subspaces of V, then SNT is a subspace. Select one: True False

A: Since this is a true or false question we can come directly into the solution without going into the…

Q: Question 3. Are the following equalities of subspaces in P3 true or false? Justify your answer. (1)…

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Q: Question 3: Show that if Y and Z are compact subspaces of a topological space X, then Y UZ is also a…

A: Let , X  be a topological space. We have given that , Y and Z are compact subspaces of a topological…

Q: Question 2. Suppose U and W are subspaces of a vector space V with basis B and C, respectively. Show…

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Q: Theorem[4.1.10] Let X and Y be a topological spaces and assume that A C X and B CY. Then the…

A: Given: A ⊆ X, B ⊆ Y. To prove: Product topology on A × B is same the topology A × B inherits as…

Q: If X a normed space and y is a proper proper Linear subspace of X. show that y has empty interior.

A: We proved by contradiction. Suppose Y has a nonempty interior then Y contains a open ball. Then we…

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: If S and T are subspaces of V, then sNT is a subspace. Select one: O True False

A: Question 1) true  Q. If S and T are subspaces of  a  vector space VTo prove: S∩T is a subspace of…

Q: Show that the set S = {(x, y) in R2 such that x =3y } is a subspace of the vector space R2 NOTE!…

A: That's easy. Have a good day!!!

Q: Theorem 7.35. Let X and Y be topological spaces. For every y E Y, the subspace X × {y} of X × Y is…

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Q: Let  S=span(e1),  T=span(e2) and  W=span(e1+e3) be subspaces of R3.  S is orthogonal to T, T is…

A: Orthogonal Subspace

Q: Give an example of a vector space V and non-trivial subspaces X.Y. Z of V such thatV =X@Y= X@Z but Y…

A: Give an example of a vector space V and non-trivial subspaces X, Y, Z of V such that…

Q: Show that W = {(a, 0, b)la, b = R} is a subspace of R³.

A: As per guidelines we are supposed to solve the maximum first questions only .kindly post another…

Q: Question 4. If U is a subspace of R", show that U-l = U. : U.

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Q: If u and v are vectors in R and W is a subspace of R with (u + V) belong to W, then both u and v…

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Q: Q. Show that (a,b,c) where c=a+b is subspace of R

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Q: 1. Is the subset V {(x, y, z, w), y = 3z, w = 2x + 1) a subspace of R*? Show the details of %3D your…

A:

Q: Question#15 Let U = {a, b}, U2 = (c,d} be subspaces of vector space V. Is U, = {a + c,a + d,b+ c,b +…

A: If W is any non-empty subset of V, then this W is a subspace of V ifiw1+w2∈W ∀w1,w2∈Wiiαw1∈W∀α∈F and…

Q: Let W = {(" ):a = 2c = b = –3d} be a subspace of M22. Then dimension of W is equal to: None of the…

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Q: If W is a subspace of R" and if u is in both W and w then u must be a zero vector.

A: I am going to solve the given problem by using the orthogonal property of a vector.

Q: If S and T are subspaces of V, then SNT is a subspace. Select one: O True O False

A: Then 0 or additive identity belongs to S and T  Therefore 0 ∈S∩T so that S∩T =∅ Let x, y ∈ S ∩ T…

Q: Consider X = {1, 2, 3, 4, 5} where the topology is {UC X|1 € U}. Find a subspace of X which is…

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Q: Question 3:- Let W be the subspace of R* spanned by the set of vectors 2 1 1 -2 3 S = 3 Find a basis…

A: We have a set of vectors S=11-21,0101,1-200,23-43 Writing this in the form of a matrix:…

Q: = E In > 1} be viewed as a subspace of R. What is the subspace topology on X? A. Discrete B.…

A: Let, y=1n: n≥1 be viewed as a subspace of ℝ.

Q: 2 2 , find the closest point to v in the subspace W spanned by 6 and 3 Given i = 6. 1 12

A: To find  The closest point to V in the subspace spaneed by the given vectors.

Q: a) Prove that W = {(a,b, 2a + b)} is a subspace of R. (more space on the next %3D if needed)

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Q: Question 2. Are the following subsets of vector spaces subspaces? Give proofs of your answers by…

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Q: Question 1 Let W be the set of all vectors of the form (x, y, z) where y=x+z+ 3. Determine whether W…

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Q: If B is basis for (x, t) then By = {BnY/BE basis for the subspace topology Ty on y IS 9

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Q: Theorem[4.1.10] Let X and Y be a topological spaces and assume that A E X and B C Y. Then the…

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Q: Question 3. Let (X,r) be a topological space and YCX. Show that if the subspace (Y, r,) is…

A:

Q: If V is a 13 dimensional subspace of R20 then the dimension of V- must be

A: Definition of Orthogonal Complements Let V be a subspace of ℝn. Its orthogonal complement is the…

Q: Question 3:- Let W be the subspace of R* spanned by the set of vectors 2 3 1 S = -2 1 -2 -4 3 Find a…

A: Let W be the subspace of R4 spanned by the set of vectorsS=11-21 , 0101 , 1-200 , 23-43

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.