Q2: Consider the following normed spaces (C([a,b]). Il-ll), (C¹([a, b]). lll.) and (C¹ ([a,b)).ll'lla). Define a linear map By :: C¹([a,b]) C([a,b]) dx is not continuous. Prove that the linear map d dx 1 d == dx (C¹([a, b]). llll) (C([a,b]). lll.)

Q: 1. Let V, W be finite-dimensional vector spaces. Define the map from L(V, W) to L(W', V') by $(T)=T…

A: Given that V and W are finite dimensional vector spaces. Let dim V=m and dim W=n. Then we can write…

Q: H is a Hilbert space, A: H → H is linear ar = for all r, y e H. Use th closed graph theorem to show…

A: For two normed spaces V and W, a linear operator T:V→W is closed if for each xn∈V, xn→x and Txn→y…

Q: Let G = <x, y | x8 = y2 = e, yxyx3 = e>. Show that |G| ≤16. Assumingthat |G| = 16, find the…

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Q: D: Remind that the set F of all functions from R to R is a vector space. 3. Prove that F = 0 E

A: Here the set F of all functions from ℝ to ℝ is a vector space. It has to be proved that F=O⊕E.

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A: A matric space X is separable if it has a countable dense subset.

Q: For a polynomial p ao + a₁x + a₂x² + + anx, let p denote the polynomial obtained from p by removing…

A: Note:  As you requested only Part C(I) so I answered here only that.

Q: 2. Are the following sets real vector spaces? Justify your answers.

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Q: Show that dim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2), where W1 + W2 just denotes the span of W1…

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Q: 10. Suppose that U and V are finite dimensional vector spaces and that F:VU is a linear map. prove…

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Q: Consider the vector space R³ and d₂(x, y) = (-1lx - y²). Show that d₂ satisfy M3 (The tringle…

A: Consider the vector space We  have  to prove triangle  inequality   that  is  we  have to how that…

Q: Let X and Y be topological spaces and endow X×Y and Y×X with the respective product topologies. Show…

A: To Show : Let X and Y be topological spaces and endow X×Y and Y×X with the respective product…

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Q2: Consider the following normed spaces (C([a, b]), ll·ll.), (C¹([a, b]), Il-ll..) and (C¹([a,b]),…

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Q: Please solve the question with a good explanation. It is Linear Algebra

A: Given:

Q: Consider arbitrary linear maps S: VW, T: WU and ToS: VU, where U, V and W are finite-dimensional…

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

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Q: (K", || |,) is called the LP-norm on K". is a normed linear space for 1 <PS00. || |,

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Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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Q: V be a vector space and T: V → V be a linear map such that T² = T. Show that img(T – I) ≤ kerT.…

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Q: Let T: R -→ R° be linear map such that T(1, y, z, w) = (y – 2x, y + w, z), then ker T A) {(t, 2t,…

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Q: Is the set V = {(x, y)|x > 0} where (x1, Y1) O (x2, Y2) = (x1x2, Y1 + Y2) and c O (x, y) = (x°, cy)…

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Q: Is a subset of the plane {(x1, x2); x1 > 0, x1x2 ≥ 1} convex? Justify.

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Q: b) S : M3,3 → M3,3 defined by S(A) = –A"? (i.e. negative transpose)

A: This is a question of linear transformation. The problem is to verify whether the transformation is…

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Q: Let H be the plane 2 X- 2y- z=0 in R', that is, H={(x, y, z)eR³ | 2x- 2y- z=0} and let F be the…

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Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

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Q: Let 7,V be a projection af the Vect or space. Accordling to this: Show that every Xe Im (T) for…

A: The given problem is related with projection. Given that T is a projection of the vector space V .…

Q: Let V and W be normed vector spaces and let f : V →W be differentiable at a e V. Let h e V. In each…

A: The mapping f:V→W will be linear , if for any scalar k  a,b∈VFa+b=Fa+FbFka=kFa For any two elements…

Q: Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact.…

A: Given: X is a finite dimensional normed space and A : X → X is a linear map. We have to show that A…

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Q: Let P(V) be a projective space of dimension > 2, and let [v], [v2), [v3] € non-colinear points…

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Q: Consider linear vector space c¹ ([a,b]) = {f: [a,b] → Rf and f are continous on [a, b]} ||f||0,1:=…

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Q: If X is finite dimensional hor med space o T: X-Dy is a Linear operator from X to normed space y.…

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Q: Find the distance from y to the plane in R³ spanned by the set (U₁, U₂} 6 - 2 4 y = -9 a. b. C. 43₁…

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Q: (b) Let X bę a normed linear space. Let Вх, г) %3D (у є Х: || x-у || <r). Show that B(x, r) = {y e…

A: To show - The ball B¯(x, r) is closed. Indeed, for any given y ∈ M \ B¯(x, r) we set R = d(x, y)…

Q: Consider these lines in three-space. ₁: [x, y, z]= [5, -3, 6] + [-4, 1, -2] l₂: [x, y, z] = [3, -8,…

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