Prove that the dual space of 10 is not isometric to 11 but contains a subspace isometric to l1.

Q: Let X₁ be a closed subspace and X₂ be a finite dimensional subspace of a normed space X. Prove that…

A: Here we have given that X1 be a closed subspace of a normed space X. So, the quotient space XX1 is…

Q: 5 A subspace W CR' such that W and W- are isomorphic.

A: Given that W is a subspace of ℝ5 let W⊥ is the orthogonal complement of W To verify whether W is…

Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

Q: Show that W is a subspace of R¹.

A:

Q: 2. Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Given W is an n-1 dimensional subspace of V which is n-dimensional. To show V has a basis B…

Q: does a basis for a subspace W always span W

A: our objective is to conclude.

Q: Suppose X is a topological space whose topology is coherent with a family B of subspaces.  Prove…

A: Given X is a topological space, whose topology is coherent with a family B of subspace.Using the…

Q: A subspace U of R^2 is a subset which satis es three properties: A) 0∈U, B) U is closed under…

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Q: Verify that if W is a subspace of an inner product space V and v is in V, then perpw(v) is…

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Q: Give an example of a subset (not subspace) of R3 that has an infinite number of elements, is closed…

A: A subset W of vector space V over the scalar field K is a subspace of V if and only if following…

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

A:

Q: Construct the vector space of all 3-tupies over GF(3). Form a two-dimensional subspace and its dual…

A: Given, the vector space of all 3-tuples over GF(3).

Q: Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on…

A:

Q: 2. Let W be an (n − 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Please find the answer in next step

Q: The empty subset of any vector space V is not a subspace of V. True False

A: The empty subset of any vector space V is not a subspace of V The given statement is false.

Q: Find a basis for the subspace of R3 spanned by

A: In this question, we find the basis for the span of subspace S. S={(-1,2,5),(3,03), (5,1,8)}

Q: Prove that the set ? = {(a + b, a -b): a and b are real numbers} is a subspace of ℝ2 with the…

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Q: Prove that the subspaces of R3 are {0}, R3 itself, and any line or plane passing through the origin.

A: Let S be a subspace for R3   Hence,              If S is the subspace then by definition:…

Q: Let W1 and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Consider W1 and W2 be the subspaces of finite dimensional vector spaces V.

Q: Let V be a finite-dimensional inner product space, let W be a subspace of V, and let projw : V → V…

A: Please check step 2 for the solution.

Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

A:

Q: Let S be a subset of a vector space V. Show that span (S) is the smallest subspace containing all…

A: Given that, S be a subset of the vector space V. We have to show that spanS is the smallest subspace…

Q: Find the orthogonal projection of 5 16 onto the subspace W of R spanned by 4 -4 and 24 projw (7)

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Q: Let W be a subset of the vectors in R where W = X2 X4 -x3 = x, - x, }. Show that W is a X3 3. X4…

A: We will use the formal definition of a subspace of a vector space to prove that, W is a subspace of…

Q: Find the orthogonal projection of onto the subspace W of R' spanned by -1 -1 -1 1 1 projw (7)

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Q: Let X = c, the space of all conver gent sequenes and M == (xn) such that , = 0 then M is not…

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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: 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: If S is a set of vectors in an inner product space V(F) then prove that S- is a subspace of V.

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Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

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Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

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Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

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Q: Prove that the dual of 1 is not isometric to 1¹, but contains a proper subspace that is isometric to…

A:

Q: Show that the set W = {(x1, 0, x3): x1 and x3 are real numbers} is a subspace of R3 with the…

A: Given: the set W = {(x1, 0, x3): x1 and x3 are real numbers} is a subspace of R3 with the standard…

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: that set of of degree subspace of P₂ Prove or disprove all Spolynomials is 43 not? Or

A: P2 is vector space of real polynomial of degree less or equal to 2. That is P2=px=a0+a1x+a2x2:…

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: Prove that p2 is a subspace of p3.

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Q: If V is any three-dimensional subspace of R°, then V has infinitely many bases.

A:

Q: If X is an inner product space, and A is subspace of X then A = A+ True False

A: Let A be a subspace of an inner product space X. Then (i) A⊥ is a subspace of X ; (ii) A∩ A⊥ = (0);…

Q: Determine whether the following set W = {(a1, a2, a3) E R° : a1 3az and az = -a1} %3D is a subspace…

A: The given set is W=a1,a2,a3∈ℝ3:a1=3a2 and a3=-a1 Let, A, B∈W and α∈ℝ So, A=(a1,a2,a3), a1=3a2 &…

Q: Find the dimension of the subspace Hof R2 spanned by

A:

Q: Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying BnW = Ø. |

A: This question is related to vector space . Solution of this question is

Q: If X is an inner product space, and A is subspace of X then A = AL %3D O True O False

A: TRUE

Q: Let W be the subspace spanned by Note that this basis is not orthogon Tind the orthogonal projection…

A: We have to find the orthogonal projection of into W.

Q: Let X be complete metric space, then a subspace M of X is itself complete iff the set M is closed in…

A: Hint: First suppose that M is compete then show that M is closed then suppose that M is closed and…

Q: Prove that is U1 and U2 are subspaces of V then dim(U1 + U2) dim(U1)+dim(U2)-dim(U1 N U2). Using…

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Q: Let W, and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Given that  W1  and W2 are the sub spaces of a finite dimensional vector space V.

Q: A subspace U of R2 is a subset which satisfies three properties: A) 0 EU, B) U is closed under…

A: