10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

Q: 2. An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two…

A: We know that, The sum of distances to any point on the ellipse x,y from the two foci c,0 and -c,0 is…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

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: 10. Prove that T, is topologically conjugate to the quadratic map F(z) = 4z(1 - z).

A: Solution of 10. To prove that T2 is topologically conjugate to the quadratic map F4, we have to find…

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

A:

Q: If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

A: We will prove the given statement.

Q: (1.) Prove that (R", d) with d : R" × R" → R defined by d(r, y) = E(Ik – yk)² k=1 is a metric space.…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Prove that L∞[a, b] is complete with respect to the norm || · ||∞.

A: The elements of L∞[a,b] are the essentially bounded measurable functions.

Q: 2.11 D.ll Let cRidul be amet (ic Then the falwing ard space are equivalent @ P is continuong onX (a…

A: Given that ℝ,d be a metric space. The given map is f:X→Y. Then to show that the given statements are…

Q: ) Find the fomily of the equ ation of all lines in 3-space perpen diuler to 2x -y +2u u Given line

A: Family of lines contains parameters.

Q: Show that every plane through the origin in R³ may be identified with the null space of an element…

A:

Q: Find the set of ordered pairs (T0, Yo) for which Picard's Theorem guarantees that the IVP y ry %3D y…

A: Given,            y'=xyx+y, y(x0)=y0

Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Show that R is not compact.

A:

Q: 3. Find all the fixed points and classify cach of them: f(r,u) ryev,

A: Solution:- Fx,y=xye-x2-y2 Put Fx=0 and Fy=0 Fx=e-x2-y2y-2x2=0and Fy=y-2y2 e-x2-y2=0 So,  0,0,12,12,…

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

A:

Q: 4:x→X (when X is a normed Space) is continous at xo € X iff for any Sequeue ie AS n s

A: Let f:x→X (when X is a normed space) is continuous at x0 ∈X iff for any sequence  xn ∈X, xn→х0  ⇒…

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

A:

Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: Let, T:R' → R';T(x,y,z) = (x + y+ z,2y+z,2y+3z) 2. Find the dimension of KerT and ImT

A:

Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

A:

Q: snip

A: Inner product of two vectors is equal to sum of product of corresponding elements of the two…

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

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

A:

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

A: Solution:-

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: 2. What is the largest possible c in (1) if X=R² and x₁=(1,0), x₂ = (0, 1)? If X = R³ and x₁ = (1,…

A: Given: c To find: a) X=R2b) X=R3

Q: 2. Explain why there does not exist a measure Space (X,S, M) with the pro porty %3D

A:

Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Q: . Show that if S is a subbasis for topologies T1 and T2 on a space X, then T1 = T2.

A:

Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

A:

Q: 9. verify that lx1l= max ilt)| defines. telasb] a on the Space Ccla, b]. morm Solution :-

A:

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

A:

Q: 3. (a) Show that under the mapping w = 1/z, all circles and straight lines in the =-plane are…

A: (a) We have to show under the mapping w = 1/z, all circles and straight lines in the z plane are…

Q: 1- Suppose that ,p) is a meric space, an that E CX. Then the interior E is open (m X) and the…

A: The given question is related to metric spaces. The solution is explained below.

Q: In general, for a a) S is compact 2) Sis closed at S is compact

A: We use definition of compact metric space, totally bounded set, Closed set, accumulation point or…

Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

A:

Q: = 1, prove that 1. If a + u bě+ u Ć +U %3D ди ду du ди du - 2| х +y Əz

A:

Q: Please solve this linear algebra question. Thank you.

A:

Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

A:

Q: Suppose the matri x A is 27 X 36 and the rank of A is 14. Find the nullity of A a e dimension of…

A:

Q: Let F1 and F2 be 4-dimensional flats in R6, and suppose that F, N F, # Ø. What are the possible…

A:

Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

A:

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

A:

Q: Show that the real number R with d(x,y)=|x-2y| are not a metric space

A: