. GNen that tre map D: Zs Z a ring homormarpnism,fird kero, and 0 (Zg) clefined bu D(x) = 6x is

Q: Consider the following inner product R?. on ((T1. Yı), (T2. Y2)) = ax122 + by,Y2 If ||(1, 3)||² =…

A: We will find out the required value of a+b.

Q: For z € C, define Tz: CC by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Given that: For z∈C, Define Tz:C→C by Tzu=zu. The objective is to verify the given statement. Now…

Q: H.W. Classify the following eq.,and reduceit to the canonical form: (3)U+Uyy = 0

A: As per company norms, we are supposed to answer only first question in case of multiple questions.…

Q: 3. Compute the inner product g (v,w) of v = + x. 2 əx and w = x.-+ 2y. Əx ay' Where the line element…

A:

Q: Suppose that f, g € H(D(z,r)), 1 < m < n, f has a zero of order n at z and g has a zero of order m…

A: Given that fz1 has zeros of order n at z ⇒fz1=z1−znϕz1 where ϕz1≠0 And gz1 has zeros of order m at z…

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f + 3g, v= fg, find the Wronskian of u…

A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

Q: Find a definihen ofa duisbailiky khatin | on by ris rule : for eluments A snd B g Im, ALB if and…

A: It is given that

Q: Let xe c and ye show that <xiy= y AX is qu inmer product if A is hermition positive del iniite

A: (Solving the only one question as per our guidelines) Question: Let x∈ℂm and y∈ℂm Show that x,y=y*Ax…

Q: Q3: Prove that X, = X, = ,X; = y %3D %3D %3D are symmetries of ys y"=3y" y"2+y''3 %3|

A: Given: To prove that the  X1 = ∂∂y,   X2 = ∂∂x,  X3 = y ∂∂ x are symmetries of  y' 5 y''' = 3 y' 4…

Q: The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of…

A: We will use the basic knowledge of multivariate functions to answer this question. Note that in your…

Q: 7. Use Green's Theorem to evaluate the integral f x²y dx - xy²dy where C: is the circle x² + y² = 4…

A:

Q: Let M be an elliptic paraboloid z = y² + 2?. (A) "Compute the Gauss map and the shape operator. (B)…

A: Answer for sub question a: Given x=y2+z2 A parametrization is given by: Xu,v=u2+v2,u,v Then,…

Q: Evaluate the integral ∮C 4x3y dx + x4 dy for any closed path C.

A:

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

A:

Q: Prove that if Y:nz+P(x- mz) So thdz

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Q: Drag the postulate or theorem that proves the trieng AZWX AY XW a. ATSU AXVW # SSS E SAS

A: which criteria follow to congruent triangle

Q: Consider P2 with the inner product (p, q) = | p(x)q(x) dx . Let U = span{1,2x + 2} and g(x) = 3x? +…

A: Consider P2 (polynomials of degree atmost 2) with the inner product defined as follows, p, q=∫-11…

Q: Consider P2 with the inner product (p, q) = | | p(x)q(x) dx . Let U = span{1, 2x + 2} and g(x) = 3x2…

A: See the detailed solution below.

Q: Let C:z|=1 in the counterclockwise direction and I = ¢. Lnzdz, J = $. zdz, and K = $. tanzdz. For…

A:

Q: (2 be equipped with the inner product (x, y) = XkYk: Let T : H → H be defined by |0 (Tx)k : if k =…

A: Given H=l2 is equipped with the inner product x,y=∑k=1∞xkyk and T:H→H be defined by  Txk=0 ,…

Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

A:

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

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Q: The orthogonal projection of y onto u is the same as the orthogonal projection of y onto cu whenever…

A: solution

Q: 3. Use Green's Theorem to evaluate G³ï+xj)•dr around the perimeter of the rectangle 0<x<2,0<y<3.…

A: Given ∫C(y2i+xj).dr We know that dr = dxi+dyj Thus ∫C(y2i+xj).dr = ∫C(y2i+xj).(dxi+dyj)  =…

Q: Evaluate | CurlF nds F(x,v,z) = - yzi + xzj – xyk, and S is that part of the cylinder 4= x² + y²…

A: Given function is  Fx,y,z=yzi+xzj-xyk To evaluate ∬SCurlF·nds where S is the part of the cylinder…

Q: Consider a dihedral groop Dy=<agb: asb= (abj=1 Ds cuss all Dy and verify Subgooups of "Dy=…

A: See answer below.

Q: Let g, h: -R and /:R'+ R be defined by /(x, y) = (g(x, y), h(x, y). Assume curl /(x, y) = 7x – 7y,…

A: Let g,h:ℝ2→ℝ and f:ℝ2→ℝ2 and fx,y=g(x,y),h(x,y). Assume that ∇×f=7x-7y, ∇f=7y-8x and…

Q: Let D be the portion of z = 2x + 5y inside x² + y² = 1, oriented up, and let F = (3, 2, -2). Compute…

A:

Q: Let M be an elliptic paraboloid a = y? + z². Compute the Gauss map and the shape operator. Find k…

A:

Q: Let C:z-1 in the counterclockwise direction and I= $dz, J = . Lnzdz, and z+5L K = 6 tanzdz. For…

A: Cauchy Gourast theorem: Suppose that function 'f' is analytic in simply connected domain D, then…

Q: Let C:z-1 in the counterclockwise direction and / = . Lnzdz, J = , zdz, and K = $. cotzdz. For which…

A:

Q: bound the module of the contour integral (image) where C= { z E C : |z - 1| = 2}}

A: Introduction: Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement…

Q: The integral (r+2y)e-dedy can be written and evaluated after using the transformation u = r+2y and v…

A:

Q: 5. Let M(z) and N = {z € C: 0 < arg z <} and D(0,1) = {z € C: |z| < 1}. Prove z2+i a) M(z) is a…

A: We are given the map, M(z)=z2-iz2+i..........(1) and two regions We need to prove two things,

Q: For z ∈C, define Tz: C →C by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Firstly, find out Tz*.

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: prove that the Lollowing examples satisly Laplace eqadion are -2y Cos 2X

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Q: Suppose {V1, V2}is an orthonormal set. Let w = 301 – 252 and z = 201 + 02. %3D Find w· Z, the dot…

A: v1,v2 is an orthonormal set.w=3v1-2v2 and z=2v1+v2

Q: a. Show that T: 1² → 1² defined by Tx = y = (n), n, = r, is compact where x = (5,) and where0 <r< 1.

A: Show that T: l2→l2 defined by Tx=y=ηj, ηj=rjξj is compact where x=ξj and 0&lt;r&lt;1. Definition:…

Q: 5. Find the curl of U = e*yax + sin(xy) a, + cos² (xz) az. A. 2z cos(xz) sin(xz) a, + [x cos(xy) –…

A: FORMULA USED- For  f¯=pax+qay+razcurl f=axayaz∂∂x∂∂y∂∂zpqr

Q: 2. For The Lineor mapping T:RT→R deffined by FLXys,t) = (x-ytstt,' xtas-t, xty+3s-3t) find W OPE a…

A:

Q: Let C:z=1 in the counterclockwise direction and I = . dz, ] = f. Lnzdz, and z+5 K = $, tanzdz. For…

A: Here we will check whether the function is analytic on given region, And then will decide the…

Q: O let T:R?→R² be a lineas tanafor- mation Such that T([:])- [ ]. T %3D Find matnie A Such that T(x)=…

A: Given that T:ℝ2→ℝ2 such thatT11=41  and T01=32 We have to find the matrix A such that Tx=Ax   ,…

Q: Find the angle between x and x4 in C[0,1] with its standard inner product.

A:

Q: (c) Evaluate d Gosh: where C: |z| = 2 by using Cauchy's residue theorem.

A: Given integral is ∮C dzcoshz;   C : z=2 As, we know  If z = x+ iy then z =x2+y2 As given that  z=2…

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: Let xe C and ye show that <xiy= y* AX is que inmer product if A is hermitian positive def iwite

A: Let x∈ℂm and y∈ℂm To prove that x,y=y*Ax is an inner product if A is Hermitian positive definite.…

Q: By using the Residue theorem, compute the integral e dz, where I is the circle |z| = 3 traversed…

A:

Q: Let C:/z|=1 in the counterclockwise direction and I = $ Lnzdz, J = $. zdz, and K = $. tanzdz. For…

A: