Prove that the intersection W n U of two subspaces of a vector space V is a subspace of V and prove the union W U U is not always a subspace of V.

Q: Let V be an inner product space and u a fixed vector in V . Prove that the set of all vectors in V…

A: The objective is  to prove that the set of all vectors in V that are orthogonal to u is a subspace…

Q: Prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and, whenever a ∈…

A: We need to prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and,…

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 U and W be subspaces of a vector space V. 1. Prove that UnW is also a subspace of V. 2. Must U…

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Q: Let V be a finite dimensional, complex inner product space and let U be a subspace of V. Suppose v e…

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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 U and W be subspaces of V over a ficld F. Show that the union U UW is a subspace if and only if…

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Q: Let S be a subspace of R" and let S be its orthogonal complement. Prove that S- is also a subspace…

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Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U + W.…

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Q: Prove that if M is a closed subspace and N is a finite dimensional subspace of a normed space X,…

A: This is a problem of Functional Analysis.

Q: Show if W is a subspace of the vector space X, then W is convex.

A: Definition used: A set K in a vector space V is called convex if and only if  x, y belongs to K,…

Q: Let V be a vector space, F a collection of subspaces of V with the following property: If X,Y EF.…

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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 is a subspace spanned by the u's where y=322-1, u1=1101, u2=-120-1 AND u3=-1011   To Write:…

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: Is W a subspace of the vector space? W is the set of all matrices in M, with zero determinants. n,n…

A: Use subspace test if u and v in W them au+v must be in W where a is any scalar

Q: Prove that the set of vectors of the form (a, b, c) where b = a +c is a subspace of R³.

A: According to the given information, it is required to prove that:

Q: Let H and G be subspaces of a vector space V. Show that HnG is a subspace of V.

A: We have to show that intersection of two subspaces is a subspace.

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: We write y = p+o, where  is the projection of y, on W and o = y −p is orthogonal to W Since  are…

Q: Let V be a finite-dimensional vector space, and let U, W be subspaces of V. Prove that (U + W)° C…

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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: In this question, the concept of the Orthogonal Decomposition theorem is applied. The vector…

Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U # W.…

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Q: Prove that if S ⊆ R is a vector subspace of R, then either S = {0} or S = R.

A: Here, if s={0} then we are done.  But suppose that S⊂ℝ and S≠{0} is a vector subspace of ℝ to prove…

Q: Let U and W be subspaces of a vector space V. Show that U ∩ W isa subspace of V and that U + W = {u…

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Q: Let H be a subspace of a vector space V, and suppose that V has dimension d. Which of the following…

A: a) Given a vector space V, let S=v1, ..., v2 be a basis for V. Consequently dimV=d.  Now given H, a…

Q: Let V be the vector space of all square sub-matrix over R. Examine whether or not W = (A: A E V, A²…

A: Let V be the vector space of all square sub-matrix over ℝ. Let , W = A / A∈V , A2 = A We need to…

Q: Let W1 and W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that…

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Q: Let A be a linearly independent subset of a vector space FV and let W = L(A) be a subspace of FV .…

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Q: Let T be a linear operator on a vector space V, and let W1,W2, . . . ,Wk be T-invariant subspaces of…

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Q: 2. Prove that the set S is a subspace of the vector space V.(5 V = R3x3 and S is the set of upper…

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Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

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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: Show that a subset W of a vector space V is a subspace of V if and only if span[W] = W.

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Q: Let U_1, ..., U_m be subspaces of some vector space V. Show that U_1 + ... + U_m is the smallest…

A: ●Additive identities:0∈ U1 ,......,0∈ Um . ●Closed under addition: u1 ,w1 ∈ U1 ,....,um  ,wm∈ Um.…

Q: If W is a subspace of ℝ4 whose dimension is 2 then is any set of 2 vector is a basis for W

A: Answer

Q: Is W a subspace of the vector space? W is the set of all matrices in Mn,n with zero determinants.

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Q: Show that the nonempty subset W of a vector space V is a subspace of V if and only if for every pair…

A: Consider a vector space V over a field F. Suppose that W is a non empty subset of V such that…

Q: Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W.

A: To show that W is a subspace of V if and only if span (W) is equal to W.Let W be the subspace of V.

Q: If W is a subspace of R" and ax is in W, where a is a nonzero scalar and x is in R". Then x is in W/…

A: We have to check

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: Let H be the set of all vectors of the form (a – 3b, b – a, a, b) where a and b are arbitrary…

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Q: 5. Let W be a subspace of the vector space V. Prove that the zero vector in V is also the zero…

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Q: Let W be a subset of the vector space V where u and v are vectors in W and c is any scalar value. If…

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Q: Let V be a finite dimensional inner product space and let U and W be subspaces of V . Note: parts…

A: Let V be a finite dimensional inner product space and let U and W be subspaces of V. (a) If dim U+…

Q: ) Do there exist subspaces S,T, and U of R³ such that S 1 T, SI U, and TiU? if so, provide an…

A: I have considered te subspace generated by orthogonal basis vector.

Q: Let H be the subset of the vector space M of symmetric matrices of order 3 such that O z+1y 2x H = y…

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

Q: Determine whether the given set SS is a subspace of the vector space V.    (2,3,4,5)

A: For subspace test , if u and v is any element of S then au+v must be in S where a is any scalar

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 vectors are y=466−1, u1=1101, u2=−120−1 and u3=−1011. Now, the formula of orthogonal…

Q: Let V and W be vector spaces with subspaces V1 and W1, respectively. If T : V --> W is linear,…

A: Given: T:V →W is a linear transformation, V1, W1 are respectively subspaces of V and W

Q: If U is a subspace of a vector space V then U is a vector space. O True O False

A: If U is a subspace of a vector space v then U is a vector space