1) Let V be a vector space, Fa collection of subspaces of V with the following property: If X, Y E F, then there exists a Z EF such that XUYCZ. Prove that UveF U is a subspace of V

Q: Let Y be a vector subspace of a normed space X. Show that the closure of Y is also a vector…

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Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ × V₂, W) of bilinear maps is a…

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Q: Let V be the vector space of all functions from R  into R; let V\index{e} be the subset of even…

A: Let V be the vector space of all functions from R into R. and                Ve  = { f(x)  :  f(-x)…

Q: Suppose V1, ..., Vn are vector spaces then does L(V1 x...x Vn, U) have the same dim with L(V1, U)…

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Q: Let U and V be finite-dimensional vector spaces. (a) Find a formula for    dim(U X V) in terms of…

A: It is given that, U and V be the finite-dimensional vector spaces. a) We have to find a formula for…

Q: 2. Let U, V be subspaces of a vector space W. (a) Show that their intersection UnV is also a…

A: (a) Intersection of Subspaces:Base Case (U ∩ V is non-empty):Since U and V are subspaces, they both…

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 V be a vector space and assume that U, W are any two proper subspaces of V and that U is not a…

A: Given that V is a vector space and U,W are any two proper subspaces of V such that U is not a subset…

Q: 2) suppose that w is a subspace of some vector space V. Let u be any vector in w and Let ✓ be a…

A: Let W be  a  subspace  of  a  vector space V over  a  field F.Then, orthogonal complement of W is…

Q: e number of element of basis for a vector space V is called elect one: a. null space of V b. span V…

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Q: Let V be a vector space with subspaces U and W. Prove multiplication that       U ⋂W is a subspace…

A: Given:V is a vector space with subspaces U and W. To Prove:U∩W is a subspace of V.

Q: Let P be the vector space of all polynomials of degree at most 3 with real coefficients, and let H =…

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Q: If V is a vector space over F of dimension 5 and U and W are subspacesof V of dimension 3, prove…

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Q: Let V be a C-vector space of dimension n, that is, dim(V) = n. Show that V is an R-vector space of…

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Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

Q: Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

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Q: subspace spanned by = X₁Y₁X₁Y2X2V₁ + 4X2Y2, and let P be the orthogonal projection of R² onto W.…

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Q: For each vector space V and subset W specified below, determine whether W is a subspace of V or not.…

A: Note: As per our guidelines, we can solve first three subparts. Please repost remaining part…

Q: 2. Prove the following result using the definition of subspace: Let (V,+,-) be a vector space over R…

A: V,+,. be a vector space over ℝ and W⊆V.i) W≠ϕ andii) For every w,u∈W and every k,m∈ℝ, ku+mw∈W.

Q: Show that {t, e'}, which is a subset of F, the vector space containing all real-valued functions, is…

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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: Let V be a vector space with subspaces U and W. Give example with v = R2 to show that U ⋃W need not…

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Q: 4. Let W be a vector subspace of a real inner product space V. Which of the following sets is not a…

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Q: Let V be finite dimensional vector space and let W c V be subspace with complimentary subspace W'.…

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Q: The necessary and sufficient condition for a non to be a subspace of V is a - empty subset W of a…

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Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

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Q: Show that V = W₁ © W₂ for the given vector space V and the given subspaces W₁ and W₂. (a) V = R³, W₁…

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Q: a et la CV be Subspace if and

A: Answer given below.

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: 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: 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: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Q: Does the set of all functions that vanish at x = 0 and x = L form a vector space? If it does,…

A: 1) Commutativity of addition: Let f(x) and g(x) be functions in V.Then, .Therefore, addition is…

Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ x V₂, W) of bilinear maps is a…

A: Please check step 2 and 3 for solution.! We will verify axioms of vector space.

Q: Show that an arbitrary linear combination of two states Vs in the vector space V with the same…

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