If we know that W is a subset of the vector space V and we know that W contains the zero vector 0, then W must be a subspace of V. True False

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: 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: If v1 and v2 are in R" and H=Gen{v1, v2}, then H is a subspace of R" Justify

A: Given that : and are in  Let Now, we  have to recall  the definition of subspace of a…

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: Let W be a subspace of a vector space V. Show that if x is in both W and W+, then x = 0.

A: Solution is given below

Q: 3. In the following either prove that the set W is a subspace of the vector space V or give a…

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Q: Let V be a 5 dimensional vector space over the set of reals. Then the dimension of a subspace of V…

A: Dimension of a subspace of a vector space

Q: Determine whether each statement is true or false. If a statement is true, give a reason or cite an…

A: We know the definition of subspace of a vector space, A non empty subset U of a vector space V is…

Q: If S is a subspace of R", then so is the orthogonal complement of S.

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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: 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: 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…

Q: Let W be the subspace spanned by the given vectors. Find a basis for W-. 1 -1 1 W1 W2 -5 -2 1

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Q: Let V be a vector space and u and w be vectors in V. Prove that {u, w} is linearly independent if…

A: Linearly independent sets: A set of vectors {v1,v2.......vn} is said to be linearly independent if…

Q: 2. Let W1 and W2 be two subspaces of V, such that dimg(W1) = dimg(W2) = 2. Suppose that the only…

A: here we can use linear combination to prove that given set is linearly independent.

Q: 20 n) Given below are subsets of different vector spaces. Determine whether or not each subset is a…

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Q: Suppose U and W are subspaces of a vector space V. Prove that if U C W, then U + W = W.

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Q: Let U1, U2 and W be the subspaces of a vector space V. Prove or give a counter example if V = U1 ⊕ W…

A: U1,U2 and W be the subspaces of a vector space V.

Q: . In the following either prove that the set W is a subspace of the vector space V or give a…

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Q: Mark each statement as True of False. The plane z = 2 + 3x - 5y is a subspace of R³. Select an…

A: We need to check whether the given plane z=2+3x-5y forms a subspace of R3 or not. This can be…

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: 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: v1 Determine if the subset of R consisting of vectors of the form where v1 - v2+ v3 – v4 + v5- - Un…

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Q: Let W be a subspace of R" and suppose that there is a vector v in R" with v = Projwv, then which of…

A: Let us solve above.

Q: If V is a vector space and W is a subset of V that contains the zero vector then W must be a…

A: Given statement If V is a vector space and W is a subset of V that contains the Zero vector and W…

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

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Q: Let v₁ = O 1 5, V₂ = No Yes -2 -7 13 and w= - 3 -6 19 Is w is in the subspace of R³ generated by v₁…

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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: 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: If each statement is true, write a complete proof of the statement. If the state- ment is false,…

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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.