Let B = {(p,q) e R² : p² + q² = 1}. (a) Show that B is NOT disconnected (b) Given that there is no continuous mappingƒ : B → R, infer that R2 is not homeomorphic to R.

Q: Let V be a finite-dimensional inner product space over C, and let S,T be linear maps V + V. (i)…

A: 1) Adjoint Map: Let V be an inner product space and T:V→V be a linear map. A map T*:V→V is called an…

Q: Let T be function from R to R defined as T (z) = (r, x), %3D Then T is not linear operator O True…

A: FALSE  the detailed explanation is as follows below:

Q: Suppose the motion of a continuous body is given by x(X, t) = (X1 +tX2)en +(X2-tX1)ez + X3e3…

A: The motion described is by linear combinations. We can write the linear combination in the form of a…

Q: Verify Green's theorem F - dř = P(r, u) dzr + Q(z, y) dy = | | {0.Q(z, y) – ,P(x, y)} d.ædy, %3D for…

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Q: Suppose f(z) is analytic in the unit disk D={|z| < 1} and |f(z)|<= 1 for |z| <1, Prove that…

A: Assume that f(z) is analytic on the unit disk D ={z: |z| < 1}.Also assume that |f(z)|≤1 for…

Q: 3. Find points at which the mapping defined by f(z) = nz+z^n (n E N) is not conformal.

A: Given :  f(z) = nz+zn,  where  n∈ℕ. To find : The points at which f is not conformal.

Q: Given the map f:R→R4defined by f(x.y.z) = (0,0,0,0) then nullity of is: O A. 4 OB 2 OC1 OD.3

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Q: In each case find the norm of the linear map L directly from the definition: (a) L : R → R with L(x)…

A: (a)  Given that the linear transformation L:ℝ→ℝ   with  Lx=-2x. The norm of the linear…

Q: Let T1: R? → R be the projection onto the first coordinate and A = {(x, y)|x > 0 or y = 0}. %3D Let…

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Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

Q: If T: V → W is a linear map between vector spaces V and W, T is one-to-one and onto, and a ba- sis…

A: Given that T:V→W is a linear map between the vector spaces V and W. Also, T is one-to-one and onto.…

Q: Given the map f:R-R+ defined by f(x.y.z) = (0,0,x.0) then nullity of f is: O A.4 O B. 3 OCO O D-2 OE…

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Q: Let F be the vector space of all functions mapping R into R. Determine whether or not the given…

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Q: Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

Q: What is a conformal mapping? Find all z where the function 22 +1 f(2) = f(z) ez is conformal.

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Q: Use the definition of the dot product a b = |a||b| cos 0 to prove the Cauchy-Schwarz Inequality: |a…

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Q: Let T:V U be a homomprphism then ker T={0}if T is one-one.

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Q: 1. Let Qe(z) = z2 + c. Prove that if c1 such that Qe is topologically conjugate to F(z) = uz(1- z)…

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Q: [Exercise T1.13.] Prove that for any map f : R → R, a source has sensitive dependence on initial…

A: To Prove that for map f:ℝ→ℝ, a source has sensitive dependence on initial conditions:Firstly,Let x0…

Q: The mapping T : R² → R defined as T(u) = ||u||

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Q: Suppose that W is a finite-dimensional vector space over F and that T:V→W is linear. Prove that…

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Q: Let X =  R x R  x R = R3 and define the metric d: R 3 x R 3 to R  by d((x1, y1, z1), (x2, y2, z2)) =…

A: Consider X=R×R×R=R3 Define the metric d:R3×R3 to R  by d((x1,y1,z1),(x2,y2,z2))=x1-x2+y1-y2+z1-z2…

Q: Exercise 9.8. Let X be a normed space, and let X = (X*)* be the bidual (or double %3D dual) of X.…

A: Given:  X is a normed space and X** = X** is the bidual (or double dual ) of X. Also, the map JX : X…

Q: The image of the line Im(z) = -2 under the mapping f(z)= i(z) is

A: Solution:

Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: Let F(x, y, z)=z²i+ 2xj+y°k and let S be the graph of z = 4-x² -y², z 20 oriented counterclockwise.…

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Q: Theorem 7.29. Suppose f : X → Y is a continuous bijection where X is compact and Y is Hausdorff.…

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Q: Let T: R? → R be the projection onto the first coordinate and A = {(x,y)|x > 0 or y = 0}. Let q be…

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Q: Let f: S¹→ S¹ be a homeomorphism.  Prove that h(f) = 0

A: Given that, Let f:S1→S1 be a homeomorphism. Therefore we have to proved that hf=0. So, consider…

Q: Compute the norms for (a)||f(1)||, (b) ||F(x)|l2 (c)||f(x)||, where f(r) = -VI-r, defined on [0, 1).

A: Given- fx =-1-x defined on [0, 1] (a) we have to calculate fx  fx=∫01fxdx        =∫01-1-xdx…

Q: Given the map f:R3 R4 defined by f(x,y,z) = (0,0,0,0) then rank of f is:

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Q: Let T : R3 → R² be a linear mapping. 2 .T Given that T and T find T

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Q: (b) Let T : V → R be a linear map. If Im(T) = {0}, show that T(T) = (T, 0) where õ e V.

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Q: (a) Let z = x + iy and w = u + iv, express u and v in terms of x and y. (b) Sketch the image of the…

A: Given: W=sin z

Q: The mapping w = z² + 1 is not conf ormal at O z = 0 O z = i O z = -i

A: We know that analytical function is not conformal at critical points. Let's find the critical points…

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

A:

Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = <x, y>z for all x…

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Q: Let f: R → R be the function which satisfies for all (z. y, z) € R° that z cos(ry) and let C C R' be…

A: Here we have to find out line integral of function f(X,Y,Z).

Q: On a vector space W of all continuous functions from R to R, define I, : W → W as I,[f(t)]= J', f(t)…

A: Yes  given function is linear transformation

Q: Which of the following maps is not linear? O A. L-R2-R3 defined by L(x,y) = (x+y,y.x)- OB. K:R R3…

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Q: Consider the spaces lP = P(N), 1 <p<00, i=0 For which pairs of (p, q), is the identity map I : x → x…

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Q: Let V be an inner product space and T: V →V be a linear map. Suppose ||T (r) || = ||r||. Prove that…

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Q: For the map F: R³ → R² defined by F(x, y, z) = (x²y + e², yz − x)

A: Given,            F(x, y, z)=x2y+ez, yz-x

Q: Consider the following linear maps T : R² → Rª and S : Rª – M(2,R) defined as follows: 2x + 3y T(…

A: The solution is below.

Q: Show that the following subsets of C[0, 1] are isometric: {f € C[0, 1] : f(1/2) > 0} {j€ C[0, 1] :…

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Q: (3) Suppose that T: V F is a linear map. Prove that if u E V such that u 4 Ker(T) then V = Ker(T)…

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Q: Let T : R? →→ R² and S : R² → R² be a linear maps such that r() - (1) (G) = (;) TO and S(( 2 1 Then…

A: Given: T: R2→R2 ,  S: R2→R2 be linear maps such that T10=11 &amp; S11=12 To determine: S∘T10

Q: Let T:R → R° be an orthogonal projection onto the plane x2 = 0. Then the kernel of the map T is…

A:

Q: Let R be the ring of continuous functions from R to R. Prove that M = {f e R : f(0) = 0} is a…

A: To show that the given subset M of R is a maximal ideal, we first show that M is a ideal of R and…