3. Let X + 0 be a topological space. a) Show that if {U: | i € I} is a (finite or infinite) open cover of X, then dim X = supier dim Uj. b) Compute the dimension of Va(Tf – T, T?T, – T?T2) CA³. c) Compute the dimension of V,(TỶ – T3,T?T, – T}T2) C P³.

Q: Find the conditions on a, b, and c so that (a, b, c) ∈ R3 belongs to the space generated by u =…

A:

Q: Let Φ be a set of isometries of a space A. Three distinct points a, b, c ∈ A we call “Φ-collinear”…

A: Sol:- (a) For any three distinct points a, b, c in E2, we can always find an isometry T that fixes a…

Q: Consider the real space R3 provided with the usual inner product. Given the elements u=(1,1,1) and…

A:

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

A:

Q: Consider the real space R3 provided with the usual inner product. Given the elements u=(1,1,1) and…

A: It is given that, v=w1+w2, so that w1 is orthogonal to element u and that the set {w2,u} is linearly…

Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1/2 and (v, w) = -2, what is…

A:

Q: Thm. Let v be an inner product space. and жуєх V then ||_ [121]. [171] further equility is hold iff…

A:

Q: A ° = A – JA

A:

Q: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

A: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

Q: Let V be an inner product space and u, v € V. Prove the following identity: ||u + v||² − ||u − v||²…

A:

Q: 2. Let M be a collection of subsets of a metric space X. We say that M has proper AUBE M whenever A,…

A: 2. Given, T is the property of A,B∈M such that A∪B∈M and A∩B≠∅. i.e. T is the property such that for…

Q: Consider the set S = {(x, y) = ZxZ| |x|+|y|=2}. List all elements of S, then plot the corresponding…

A:

Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

Q: Question 4 . Let (X, d) be a metric space and let p(r, y) is complete if and only if (X, p) is…

A:

Q: Let. f be q into medric Captinyous. marping of a medric rpare %.

A: Matric space

Q: Do the following task. a)  State formally what do we mean by being a metric space of X with metric…

A:

Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given: H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show: H is a closed subset…

Q: Given any two Euclidean spaces E and F, w ension m, for any linear map ƒ: E→F, we have Ker f = Ker…

A:

Q: Show the span {(1,0, 1), (-1, 2, 3), (0, 1, – 1)} is all of R'.

A: Let x=1, 0, 1, y=-1, 2, 3 and z=0, 1, -1. We have to show that spanx, y, z is actually the vector…

Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: 8. Suppose I is a triangle formed by placing three points lircle, two of which lie on the circle's…

A:

Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: Fix a prime number p. For x EQ - {0}, write x = p" () where a, b € Z, a and b are not divisible by…

A:

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given That : Suppose H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show : H is a…

Q: Let I2 and = in R', we have 43, () (u, v) uv:= x1y1 + 122 + 3Y3. Let ui = and Uz = Show that B =…

A: Since you have Posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: 3.4 Give the definition of a separable met rie space (S, d). 3.5 Show that the metrie space (R, d),…

A:

Q: If V is a vectorial space over R with inner product. Show that u Iv u,νεν, λε |||u + 2.v|| ≥ ||u||

A:

Q: a) ] Let W = span{(1, 1,0), (1,0, 1), (2, 1, 1), (0, 1, –1)}. i. Is (9,0,9) įn W? Justify your…

A: Given that, (i) To check if (9, 0, 9) is in W. If (9, 0, 9) can be written as linear combination of…

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let V be a 5 dimensional space, and V: V → V linear. Can we have NullT = RangeT? And if the…

A:

Q: that det S E

A: Prove that ff is an isometry as,

Q: W and V are linear spaces with dimension 2, the base of each of which is (e,, ez). If the linear…

A:

Q: Exercise 8. Assume A, B are subsets of the topological space X. a) If AC B, show Å C B. b) Show that…

A: a)

Q: Q4. Use the Preimage Theorem to show that the set M = {(x, y, z, t) = R¹ : x + y − z³ − t² = 0, x² +…

A:

Q: W and V are linear spaces with dimension 2, the base of each of which is e1, e2). If the linear…

A:

Q: Theorem (Axioms of an interior): Let (M, d) be a metric space and S,T CM. Then: (1) 0 =0 and M = M.…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: Which of the following statements are true for the set S= {v,, ..., V} in R"? 1: {v2 - v1, ..., Vk -…

A: We are given the set S=v1, v2, ..., vk in Rn. We know that,     The affine hull aff(S) of S is the…