Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac < 0 and a > 0.

Q: Refer to V=C[0,1], with the inner product given by an integral. Compute (f.g), where f(t) = 16t – 12…

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

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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: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

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Q: An inner product on R² is defined as (u, v) = (Au) · (Av), [ 3 where A : -3 (a) Is (1, –2)…

A: Orthogonal Vectors: Two vectors x and y in an inner product space are called orthogonal, if x, y=0…

Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: Evaluat..f.x.dA. wha.is.the..sgiem.in.the.fiigt. R quandrant..bounded..xy=16. X..8....y.z.. y.e.x...

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Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

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: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

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Q: Prove that C[x, y]/ is an integral domain

A: Gauss's lemma: A non-constant polynomial in ℤx is irreducible in ℤx if and only if it is both…

Q: Consider the region RR in the xyxy-plane that is described by the intersection of the four lines:…

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Q: Show that y^3dx+3xy^2dy=0 is separable and exact

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Q: 2. Let W be the union of the first and third quadrants in the xy- plane. That is, let W = xy 0 a. If…

A: According to our company gardening can answer only one question. If a specific questions are asked…

Q: Let Ps have the inner product given by evaluation at-2,-1, 0,1, and 2. Let p(t) = t2- 2t and q(t) =…

A: Let P3 have the inner product given by evaluation at -2, -1, 0, 1 and 2 . Let p(t) = t2-2t  and q(t)…

Q: According to the Divergence theorem fF.ds = , divFav Use the Divergence Theorem to find the total…

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Q: 3. Find a field F(x,y) = (M(x, y), N(x, y)) in the xy-plane with the property that at each point

A: Let f = xi + yj  Then F = ff = unit vector = xi+yjx2+y2

Q: Suppose F (x, y) = (2xe-", 2y – a²e="} (a) Find a scalar function f (x, y) such that F = Vf (b) Use…

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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: Find the potential function f for the field F. 6..6. F=7x°y°z' i+ 6x y°z'j+ 7x'y°z°k O A. f(x,y.z) =…

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Q: Let xbe inner Product and ||x1|=|ly|| = 1 x+y Lix.y ? why?

A: Given that X is an real inner product space. x ,y be two unit vector from X then need to show that…

Q: Consider: J²u J²u + = 0, əx² Əy² ди ər u(x,0) = 0, (PDE) (BC1) (BC2) (BC3) u(x, y) remains bounded…

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Q: Show that the quotients field of the integral domain Z[i]={a + ib: a, b E Z} is K= {x + iy: x, y E…

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Q: Evaluate. where G is a solid that lies in the first octant bounded by x^2 + y^2 = 3z and z = 3.

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Q: Is = xjx + yjy+zźz3 an inner product on R'?

A: No it is not an inner product.  Inner product looks like a dot product.

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

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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: Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizable

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Q: Given that the vector field F(x, y) = (2x-3y, 2y-3x) is conservative in the xy-plane. If (x, y) is a…

A: We have to find the value

Q: (b) Perform the Gram-Schmidt orthogonalization process for u = x +1, u2 = x? + 3, u3 Va using the…

A: b. Given that               u1=x+1, u2=x2+3, u3=x Using Gram-Schmidt orthogonalization process for…

Q: (10) Given that the vector field F= 2z i- 3 x j + 4 y k Evaluate f F. dr where S is the portion of…

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Q: Verify Green's Theorem by evaluating both integrals aN aM y2 dx + x2 dA ду ax for the given path. C:…

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Q: The space is CJ0,2r] and the inner product is (f.g) = J r9gt) dt. Show that sin mt and sin nt are…

A: Given m≠n∫02πsinmtsinntdt=12∫02πcosm-nt-cosm+ntdt=12sin(m-n)tm-n-sin(m+n)tm+n02π=0

Q: 2. Let i = y?x + xŷ + z²ż, and let S be the paraboloid z = x2 + y? with 0 < z < 1. Show that fci. de…

A: Given  v→=y2x^+xy^+z2z^ and S be the paraboloid defined by  z=x2+y2      , 0≤z≤1 The boundary C of…

Q: Consider P2 with the inner product = f (x)g(x) dr. Then d(x, 2) Jo

A: d(x,2) represents distance between the vectors x and 2 P2 is the set of all polynomials with degree…

Q: which of the sets is orthvoonal under the given inner product on C[0, 1] ? = S furgex) dx I. {1,…

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Q: Apply Gauss Divergence theorem for F = x²i + y²j + z²k_over the cuboid S bounded by 0 < x< 1, 0< y<…

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Q: 3) Evaluate curl(F F(x, y. =) = (y=+2xy, 2xy + x + =, xy + 23z) to show that is conservative.

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Q: 7. Let S be the surface of a solid cube with domain D = {1 < z < 2, 1 < y < 2,1 < z< 2}. For field F…

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Q: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: Let r; = ro + hj for j = 0,1,2. Use the quadratic polynomial interpolating f at those nodes %3D %3D…

A: The solution is given below

Q: $ ((x,y) (E,6) )e product Prove 2 xt + 3yt on e?. Gompute inner IL(,2), where the norm i's…

A: We have to prove that f( (x, y) , (t, t) ) = 2xt + 3yt is an inner product on R2 . 1.) &lt; ( 0, 0 )…

Q: [0, 1]. If f(1), g () e V then show that (f(t), g (t)) = [ƒ(t) g (f) dt defines an inner product on…

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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: Use the given transformation to evaluate the integral. IL 3(x + y) e- da, where R is the rectangie…

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Q: (b) Perform the Gram-Schmidt orthogonalization process for u = x +1, u2 = x² + 3, u3 = Va using the…

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Q: Justify the following statement: The flux of F = (x3, y3, z3) through every closed surface is…

A: The flux through a closed surface is given by Gauss's divergence theorem, Flux, ∫∫∫∇→·F→dV=∯F→·n^ds…