6. Consider FV, where V consists of all real-valued functions. Let W C V, where W = {f € V | |f(x)| < M, Væ € R} i.e. W consists of all bounded functions. Show that W is a subspace of V.

Q: If W is a subspace in an inner product space V and V = W + W, prove that V = WW¹.

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Q: 1. Let e Z(X) and let A = {xe X: f(x)=x}. Show that M is a subspace of X.

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Q: 3. Let W = {(x,y,z): x? + y? = z}. Is W a subspace of V = R3? %3D

A: We need to check if W is a subspace or not.

Q: 15. Let U and V be subspaces of a vector space W. Show that if W = UV, then UnV = {0}.

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Q: Show that {f in C[a,b]:f(a)=f(b)} is a subspace of C[a,b].

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Q: 1. Let f: R" → R be the function f(x) = cx+b for some fixed vector c ER" and scalar b. Show that f…

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Q: show thal the set of all poiuts in the Subspace of IR 3. plane ax +by+CZ =0 is +cz = ois CA

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Q: Let V = R, the real line. Let W be a subset of V, where W = {x € R|x ≥ 0}. Is W a subspace of V?…

A: Given that the vector space V=R , the real line  . The subset W of V is said to be a subspace of V…

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

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

A: We will use basic knowledge of dual of normed linear spaces and the concept of isometric…

Q: 1. Show, for every v e V, || projw v|| < ||v||. 2. Define W+ and show that WOw+= {0}.

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Q: Let V be the vector space of all polynomials in the variable x over R. Show that the mappings…

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Q: Suppose U = {(x, x, y, y) = F¹: x, y € F}. Find a subspace W of F4 such that F4 = UW.

A: Our objective is given below:

Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

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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: b.) show that IR³ - W₁ + W₂. t C.) Find a subspace V such that IR³ =W₁ U.

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Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

Q: (b) Consider the 2nd order linear differential equation xy" + y = 0. (i) Given that y₁ = ln x solves…

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Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 5tj + 2tk 5 ** 11 0 Z 2 2 Interval [0,…

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Q: Example Let f be a Lipschitz function on [a, b]. Then f is of bounded variation of [a, b], and TV(f)…

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Q: Is the set W = {f(x) = P(F) : ƒ(x) = 0 or f(x) has degree n} a subspace of P(F) if n ≥ 1? Justify…

A: Set : W= fx∈PF : fx=0 or fx has degree n

Q: 3. This problem is about set addition and subspaces. Let d = and dz = and V = span{d}, %3D U =…

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Q: 1. Let V = C[-2,2], the vector space of continuous functions on the closed interval [-2, 2]. Let f,…

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Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V…

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Q: 1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2). Let f,…

A: We can just multiply h and k and integrate  Then after simplifying we are getting answer

Q: Determine which (if any) of the following sets U, V,W are subspaces of R". For those that are,…

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Q: Find the orthogonal projection of onto the subspace W of R4 spanned by projw() = --B 7 010

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

A: Vector space.

Q: D: Remind that the set F of all functions from R to R is a vector space. 2. Prove that for any…

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Q: Let P denote the set of all functions f on [0, 1] such that there exists at least one point x E [0,…

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Q: Prove that the vector space C of all continuous functions from R to R is infinite-dimensional. Shew…

A: We have to prove given statement:

Q: If W is a subspace of R" and ax is in W, where a is a nonzero scalar and x is in R". Then x is in W/…

A: We have to check

Q: Given a linear map f: E→ F, the set Im f is a subspace of F and the set Ker f is a subspace of E.…

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Q: Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, and the following applies.…

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Q: Let X be a normed linear space and let P, Q E B(X). Show that the transformation T: B(X) B(X)…

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