Prove that W₁+W2 is a subspace of V that contains both W₁ and W₂. Prove that any subspace of V that contains both W₁ and W₂ must also contain W₁ + W₂.

Q: Suppose U, W,, W, are subspaces of a vector space V. Show that (Un W,) + (UnW,)Cun (W, + W,) Find…

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Q: If V3(R) is a vector space and W₁ = {(a,0, c): a, c = R} and W₂ = {(0,b,c): b, c = R} are two…

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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: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

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Q: If W₁ and W₂ are subspace of a vector space V(F), then show that W₁ + W₂ is also a subspace of V(F).

A: W1 and W2 are the subspace of a vector space BF We have to show W1+W2 is also a subspace of BF

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: f W1 and W2 are subspace of a vector space V(F),  then show that W1+W2 is also a subspace of V(F)

A: Given that W1 and W2 are two subspace of a vector space V over a field F.

Q: Please go through each option with explanation

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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 and state its dimension for the following subspace in Rª 3r+ 6s-t 6r-28-2t -9r+ 5s + 3t…

A: Given that 3r+6s-t6r-2s-2t-9r+5s+3t-3r+t: where r,s,t∈ℝ

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: If V3(R) is a vector space and W₁ = {(a,0, c): a, c = R} and W₂ = {(0, b, c): b, c = R} are two…

A: Provided that V3 is three dimensional vector space over field R and V has two W1 and W2 subspace…

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 W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector…

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Q: If we know that W is a subset of the vector space V and we know that W contains the zero vector 0,…

A: This statement is False.  By using property of subspace we can prove this statement is False.

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

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Q: Let M3,3 denote the vector space of all 3 × 3 matrices. Let L3,3 denote the set of all lower…

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Q: Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that v1, V2, V3…

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Q: Determine if v is in the subspace of P3 given by span{x + x - 5, x2 + 2x + 1, x-x2 + x}. v = x? + 2x…

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Q: Suppose W is the subspace of R“ satisfying the equation x1 + x2 + x3 + x4 : A. Find a basis for W-.…

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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: 2а + b - Зс 4а — 2b + 5с (а) S %— E R* : a, b, c ER a а +b+c a X1 X2 (b) S = ER: 2x + х2 — 7хз + 8х4…

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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: S� is a set of vectors in a vector space V�, then SpanS is a subspace of V�. Please show that this…

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Q: how that if T is a linear operator on V and W1, W2 are T-invariant subspaces of V, en W1+ W2 is a…

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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: Explain why the subset а — b S = a +b а,bER -26 – 3a of R3 is or is not a subspace of R³.

A: Given subset of R3 is  S={a-ba+b-2b-3a|a,b∈R} explain if S is subspace of R3

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…

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Q: Show that P2 is a subspace of P3 (See polynomial vector spaces.) the definition of the

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