b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U of V such that U N null T = {0} and ran T = {Tu | u €U}. W oro both finite dimensional, Prove that there exists

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A:

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: 3. Determine which of the following sets of functions are subspaces of F[a, b]. a) All functions f…

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Q: If X, is a dense subspace of a normed linear space X show that X X'.

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A:

Q: 2) One of the following subsets of IR2 is not subspace: a) S = {, xeR} b) s = {], = x3} %3D %3D e) s…

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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: - If X is a vector space and if X= MN, where M and N are subspaces, prove that N is isomorphic to…

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

A: Given that V is a finite dimensional vector space and U is subspace of V such that dimU=dimV. Let…

Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

Q: Q7) Let V is a subspace of R", with an orthognal basis|v1,V2,...,Vp. a) Prove that Proj,v,=v1 b)…

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Q: 1 for which x value span the vector subspace? -1 13 - 2 -1 O A) 2 в) .1 O C) 0 D) 1 O E) -2

A: We have to find x

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…

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A:

Q: Let (R, Tus) be the real line.Let r ∈ R such that 0 ≤ r < 1 and consider I = [0, 1] as a subspace of…

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Q: II. Decide which of the following are subspaces of corresponding vector spaces: C R?

A: Vector space.

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Q: Find the orthogonal projection of V = onto the subspace W of R' spanned by U1 = , U2 Uz = VW ||

A: