Suppose that A = SS−1, where  is a diagonalmatrix with diagonal elements λ1, λ2, . . . , λn. Show that Asi = λisi, i = 1, . . . , n.

Q: In AMNP, T is the centroid. If MR=30 find MT. N R T M S

A: We are given that T is the centroid and MR=30. We know that the centroid divides each median in a…

Q: Suppose that v1,..., Um is linearly independent in V and wE V. Prove that dim span(v1 + w,..., Vm +…

A: Let u1,u2,⋯,un be a set of vectors in a vector space V. This set of vectors is said to be linearly…

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: Evaluate O xy?dx + a°dy, where C is the rectangle with vertices (0,0), (6,0), (6,3), and (0,3)

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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: If A = x a, – zy a,, evaluate § A · dī around the path L defined by 0 < x < 2,0 < y < 1 as shown in…

A: Stroke's theorem:         It is also known as Generalized stroke's theorem. It states that the…

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: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (5,0,0), (0,2,0), and (0,0,4)

A: # Given integral is      ∫∫E∫ zdv

Q: If W(-10, 4), X(-3, -1), and Y(-5, 11) dassify AWXY by its sides. Show all work to justify your…

A: Given,           If W-10, 4, X-3, -1, and Y-5, 11 of ∆WXY.

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: Given the bases B={u , u2} and B'={u'1, u'z} for R2, where , U2 and u'2 = The transition mat rix…

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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: 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: In(u cos v) and y = usin v. Evaluate and z at the point 2е" In y, x 3D (u, v) = (2, 7/4). Let z = du…

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Q: [2 Lo 2. and defined T: R² → R³,by T(X) = AX. Let A Find the images under T of u = |E|, and v =

A: Given a linear transformation T:R2→R3 by T(X)=AX where A= 2002

Q: Let G(u, v) = (, uv). Use the Change of Variables Formula to compute the area of the image of [1, 4]…

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Q: Given: 2J and ZL are rig of JL and MN. Prove: AJKN E ALKM Statem 1. ZJand ZL are right a 2. Kis the…

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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: Show that the integral S. (3x²y + y)dx + (x3 + x)dy is independent of path, if it is, then evaluate…

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Q: Show that, in the triangular region with vertices (0,0), (1,0) and (1,1) the following double…

A: Double integral

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: Given the bases B = {u1, u2} and B' = {u'1, u'2} for R?, where and u'1= The transition mat rix Pg-B'…

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

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Q: . Let C be the triangle with vertices (0,0), (2,0), (0,4) directed counter-clockwise, and let F(x,…

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Q: Show that every positive sol fron of Ay + B, k=osl,2- & bosunded, %3D JK is i-e, ogy sB-if A<1. 1-A

A: Given yk+1=Ayk+B, k=0,1,2,... Then the objective is to find the solution at which the sequence is…

Q: f (lnx + y) dx − a² dy, where & is the rectangle with vertices (1, 1), 1. Use Green's Theorem to…

A: Let I =∮(ln x+y)dx-x2 dy, where the region is C which is a rectangle with vertices…

Q: Find the decomposition of a(t) a = arT + așN into tangential and normal components for r(1) = (5t,…

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Q: Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y + z+ w= 6 3x…

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Q: tnd the kes reetor projection of anto r if

A: Given : u→=5i→+2j→ and v→=6i→-3j→

Q: Given the bases B= {u,,u2} and B'= {u'1, u'2} for R, where U2 and The transition mat rix Pg-g from B…

A: We first find the coordinate vectors of u_1 and u_2 relative to basis B' ,then we will form the…

Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

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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: N is the centroid of triangle HIJ. If KN=9, IM=15 and LN=5. Find KJ K. N/

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

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Q: Find the Ker(T) and Rng(T) and their dimensions. T:R →R defined by T(x)= Ax, where 1 -1 27 -3 3 -6

A: The objective is to find ker(T) and range (T) and their dimension.

Q: If x = v'u and y = 2uv, show that JJ' - 1, where J is the jacobian of u,v with respect to x and y…

A: We have find the jacobian

Q: 4. let f(x, y) = x² – xy + y² – y. find the direction u and Duf at Po(1,–1) for which Duf is the…

A: Given:- f(x, y) = x2 -xy+y2 -y To find:- Direction u and Du f at point p0 (1, -1) for which Du f is…

Q: Use Green's Theorem to evaluate x dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

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

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Q: Let V =(1,2,3,4),V=(0,2,t,3) and w=(!,!,,1) 1. Find t so that u t v and w are to each other. or…

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Q: In AFGH, M is the centroid. If FM=20 find FK. G J K F H L

A: Given: In △FGH,  M is the centroid of the triangle. FM = 20

Q: .The primal form of SVR with soft margin is M 1 min |w+CE(6i + &;) i=1 s.t. y) – w"x1) – b < e+ §i,…

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Q: Let A e Mnxn(F) be upper triangular. Prove that det(A) is the product of the diagonal entries.

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Q: Use Green's Theorem to evaluate x' dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

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Q: Use Green's Theorem to evaluate S. axy dy + cxy dy, where C is a counterclockwise oriented…

A: Given : a=2, c=8 We have to use Green's Theorem to evaluate            ∫C axy dy+cxy dy where C is a…

Q: b) Show that the following wo matricegane orthogonal tker A =-V3 -2VTO 3/T15 15 E/NTS

A: