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## Related Algebra Q&A

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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: Either prove or disprove that the set V is a subspace V ⊂ R^n defined by V={x: Ax=λx}where A is…

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Q: Prove that a subset W of a vector space V is a subspace of V if and only if 0 EW and ax+ y E W…

A: First proving the first part: If W is the subset of V and also the subspace of V Claim : 0 W and…

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 V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U…

A: In a finite-dimensional Vector space, every subspace is finite-dimensional.

Q: Suppose W1 and W2 are subspaces of a vector space, V. Define W as follows: W = {w1 + W2 | W1 E W1…

A: As W1 is a subspace of V , hence (a) 0∈W1 (b) αw1+βw1'∈W1 for every w1 , w1'∈W1 As W2 is a subspace…

Q: 17. Prove that a subset W of a vector space V is a subspace of V if and only if W + Ø, and, whenever…

A: We Know that Let V be a vector space over the field F and let W E V . Then W will be a subspace of V…

Q: If S is closed Subspace of a normed Space (M, f), then prove that the function fo : R defined by f.…

A: If S is a closed subspace of a normed linear space M, f. Prove that the function fQ:MS→ℝ is defined…

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Q: Let X be a topological space, and let Y C X have the subspace topology. Prove that C C Y is closed…

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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: Consider the following subset of R": W = {(x1,x2, 83, L4, 25) E R° | r1 = x2 = 13, 14 + 15 = 0}. (a)…

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Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

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Q: Let β be a basis for a subspace W of an inner product space V, and let z ∈V. Prove that z∈W⊥if and…

A: Let β be the basis for a subspace W of an inner product V, and let z € V.It is required to prove…

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Q: Let W, and W2 be subspaces of a vector space V. Prove that V is the direct sum of W, and W2 if and…

A: First assume that V is the direct sum of W1 and W2 W1+W2=V and W1∩W2=0 We have to prove that each…

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Q: Let A C X. Prove that if d is a metric for the topology on X, then d[(A × A) is a metric for the…

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Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

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

A: TRUE

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Q: Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ...,…

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

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Q: 3. Prove that the set S is a subspace of the vector space V V = C(R) and S is the set of f in V such…

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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: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

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Q: If S, is a subspace of R* of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠ S2 and S2≠R4.

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Q: 1) Suppose that V is a finite dimensional vector space over R. Prove that every T E L(V) has an…

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Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

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Q: Let W be a subspace of Rn. w ∩ w⊥ = {o}

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Q: Show that the set W={(x,y,1)|x and y are real} is not subspace of R3.

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Q: 5. Let H = {x E R³| x1 + x2 + X3 = 0}. Prove that H is a subspace of R3.

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Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.…

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Q: Let X be a normed space and Y C X a linear subspace. c) Show that Y is dense in X, if and only if…

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Q: Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V,…

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Q: Suppose V is an interior multiplication space with finite dimension and W, and W2 are subspaces of…

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Q: 4. Decide if the following statement is True or False: Let U, W, and X be subspaces of a finite…

A: Given - Let U, W and X be subspace of a finite dimensional vector space V such that U ∩ W = 0 and U…

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Q: Prove that X is completely regular space if and only if it is homeomorphic to a subspace of a…

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Q: Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a…

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Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

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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: Let W be a finite-dimensional subspace of an inner product space V , and let E be the orthogonal…

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Q: Let vi # v2 E R" and define W1 Span{v2}. Use the definition to show W+n W, is a subspace of R".…

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Q: Let M be closed subspace of a Hilbert space H, and fix h e H. Then h has a unique representation as…

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Q: The following subset of R W ((a,b.c): a +b+c 0} is a subspace of R'.

A: Subset of R2 is given W=(a,b,c):a+b+c=0W is subspace if and only if α,β∈W and c∈F(field)⇒cα+β∈Wlet…

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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: 2. Check the following set -- x2 W = E R"|x1 > 0 Xm over the field of reals is a subspace of the…

A: Vector space, subspaces.

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Q: Determine a generator set for the following subspaces: (c) W = {(x, y, z, t) ∈ IR4 / x + 2y − z = 0…

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Q: Prove that if U and V are subspaces of R" and U <V then V+<U+.

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Q: Which of the followings is not a subspace of the vector space V? a) V = Mxn(R),W = {A € VIAT = A} b)…

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Q: Consider the following subset S={ [x,y,z] ∈ R3 : x=2y, 2y=−z } of R3 . Is the zero vector an…

A: Use subspace test to prove S is a subspace of R3

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Q: show that if w is a subspace of a finite - dimensional vector space V and dim (w)= dim (V) then W =…

A: In this question, we will use the fact that If dim(W) = dim(V ) = n, then a basis for W is a…

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Q: Show that if W is a subspace of a finite-dimensional vec- tor space V, then W is finite-dimensional…

A: Click to see the answer