Respond as indicated 1.) Find the norm of u=(5,6,1) with the respect to the usual inner,product (dot product ) of R 2.) IrA=; then find the norm of A with respect to the trace inner product

Q: By using the properties of dot and cross product, prove the following: c) W⃗ ×(V⃗ ×U⃗ )=(W⃗ .U⃗ )V⃗…

A: To prove :  W×(V×U)=W .U)V −(W .V)U

Q: et P E L(C") be defined by, P(z1,..., zn) = (zn, Z1,.….., Zn-1) Determine whether or not P is…

A: Let P∈L(Cn) be defined by   P(z1 , . . . , zn) = (zn , z1, . . . , zn-1)Determine whether P is…

Q: Find (a) ⟨u, v⟩, (b) ||u||, (c) ||v||, and (d) d(u, v) for the given inner product defined on Rn.u =…

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Q: (a) a = (2, 6, 0) , b = (4, –1, 1) (b) a =i-j-k, b = }i+j- 4k

A: a) given that  a=<2,6,0> , b=<4,-1,1> a→=2i+6j+0kb→=4i-j+ka→×b→=ijk2604-11…

Q: Find the projection of u along v. u = 5i + 7j − 4k, v = k

A: Given, u = 5i + 7j − 4k, v = k To find the projection of u along v.

Q: (a) Calculate (u, v), dist(u, v) and ||v|| for the following inner products: (i) Euclidean inner…

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: Find the norm of a and the unit vector u in the direction of if 1 3 x = ||*|| = 13 = = 81 -4 Y ♡

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Q: Consider (u, v) = 0 for any vectors u, v e R4. Is this an inner product? Explain why or why not. %3D

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Q: Find a norm of the function f(t) =t? – 2t in space C[0, 1], %3D where norm is given by formula…

A: Given that  f(t)=2t^2-2t C[0,1] To find the norm of the function

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: Also deduce the Jacobi identity for the cross product: u × (v × w) +w × (u × v) +v × (w x u) = 0 их

A: Introduction: We know that u×v×w=vu·w-wu·v, where u, v, and w are purely imaginary.

Q: Find the weighted Euclidean inner product (u, v) = -3v1 u1 – 3vzuz of the given vectors. u = (u, v)…

A: Given u, v=-3v1u1-3v2u2 u=-3-2, v=14

Q: Find the orthogonal projection of f onto g. Use the inner product in C[a, b] (f, 9) = (r, "rx…

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Q: 4. Consider the real vector space R?. For every x = (11,12) E R?, let ||x|| = |11|+ |r2l. a) Show…

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Q: Of the following vector, prove that the mapping is an inner product for all vectors v = Vx, Vy and u…

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Q: For the vector space M, we define the standard inner product by the 22 formula (U,V)= tr(U'V).…

A: We find cosine of the angle between the given matrices.

Q: Let v = (3, 0, 0) and w = (0, 1, −1). Determine u = v × w usingthe geometric properties of the cross…

A: Cross product: more formally called the vector product a way of multiplying two vectors and produces…

Q: Which of the followings is the vector projection of u onto v where u = -2i + 4j – V5k and v = 2i –…

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Q: Prove that if r(t) has constant length, then r(t) is orthogonal to r'(t).

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Q: Find the cross product ⃗a × b and then specifically verify that it is orthogonal to both ⃗a and b:…

A: Given data:     The first vector is: a=<a1, a2, a3>=<6, 0,-2> The second vector is:…

Q: 2. Let Ā and B be nonzero vectors. Find k e R such that B – kĀ is orthogonal to Ā – B.

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Q: Consider the real vector space M2x3(R) with the usual inner product (А, В) — tr (B'A) Given the…

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Q: (10) Let the vector space P, has the inner product (p(x), q(x)) = | P(x)q(x)dx (a) Compute ||p(x)||…

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Q: Z is the centrold of AUVW. W What is ZX if WZ = x? A. 5 O B. В. ос. 2т O D.

A: Answer is mentioned below

Q: 8) Find the cross product a x b and verify that it is orthogonal to both a and b. a = (1, 1, -1),…

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Q: Let Q be an orthogonalmatrix such that QA makes sense. Show that (QA)+ = A+QT.

A: The (Moore-Penrose) pseudoinverse of a matrix generalizes the notion of an inverse,somewhat like the…

Q: 9. f(z) = e'm/4 z %3D

A: Given: Triangle  with vertices 0,1 ,i . f(z) =eiπ4 z To find:  image of given triangle under f(z)

Q: 10) Find the cross product a x b and verify that it is orthogonal to both a and b. a =  j + 7k,…

A: Let's find.

Q: Find the norm of i and the unit vector i in the direction of a if ||7|| =

A: Solutions

Q: Use a graphing utility to find u × v, and then show that it is orthogonal to both u and v.u = 2i + j…

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Q: The projection of v onto w is v=i+j+2k, and w=i+2j+2k 3 =(-i+j+2k) а. (i + 2j +2k) 5 -(i + 2j + 2k)…

A: Projection vector of a⇀ on the b→ is given by c→=a→·b→b→b^

Q: . Find a norm on R2 for which ||(0, 1)|| = 1 = ||(1,0)|| and for which |(1, 1)|| = 0.000001. %3D…

A: The hint suggests a norm :ℝ2→ℝ defined by  a,b=Aa+b+Ba-b…1 To find the value of A and B: The norm…

Q: Let v,w ɛ R" and define A = vw². Prove that ||A||2 result true for the Frobenius norm, i.e. is…

A: Let v,w∈ℝn define A=vwT. v=x1,x2,...,xn and w=y1,y2...,yn. Then,…

Q: Use a graphing utility to find u × v, and then show that it is orthogonal to both u and v.u = 3i − j…

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Q: Find (a) ⟨u, v⟩, (b) ||u||, (c) ||v||, and (d) d(u, v) for the given inner product defined on Rn.u =…

A: u = (0, −6), v = (−1, 1), ⟨u, v⟩ = u1v1 + 2u2v2 We need to Find (a) ⟨u, v⟩, (b) ||u||, (c) ||v||,…

Q: Prove the following property of the cross product: u X (v X w) = (u*w)v - (u*v)w

A: If a, b and c are three vectors, then…

Q: b- Let X₁, X2,... X (a) Find a comp (b) Find the UM

A: Solution:   (a):

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: Find the cosine of the angle between the vectors P1(x) = 1 + 2x – x² and p2(x) = -1 – 3x + 2r² with…

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Q: Use a graphing utility to find u × v, and then show that it is orthogonal to both u and v.u = −2i +…

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Q: Let S be the parallelogram determined by the vectors b, 8. and b2 = 2 Compute the area of the image…

A: Given that S be the parallelogram determined by the vectors   The area of the parallelogram S is…

Q: Compute the Jacobian of the mapping. G(r, t) = (2re", 3 + 8e")

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Q: In Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, .…

A: Take two vectors x and y in an inner product space x=x1,…,xn and y=y1,…,yn then x,y=xTy

Q: Prove the following property of the cross product: u X kv = k(u X v)

A: Let u=u1 i+u2 j+u3 k  &  v=v1 i+v2 j+v3 k. Then, kv=kv1 i+kv2 j+kv3 k Now,…

Q: Given vectors u=(u_1,u_2) and v=(v_1,v_2) from R^2 prove whether or not the function f(u,v) =…

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Q: which of the following are inner product on V, CR) where u=(d1,de] , V = ( P1» P2) %3D 2 2 2 3) =…

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Q: Find the orthogonal projection of f onto g. Use the inner product in Ca, b] (f, 9) = ["«x»9(x) dx.…

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Q: Use the given transformation to evaluate the integral. (9x + 9y) dA, where R is the parallelogram…

A: This question is based on the concept of change of variables in double integral whose formula is…

Q: (f) Find the length of the projection of i onto ū.

A: Given