2) Let R real numbers, define < x,y >= x,Y1 +4x2y2 + 6x3Y3 for x = (x1,x2, X3) and y (y1,y2, Ya). Is < x,y > inner product or not.

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: Let V=C³ with the standard inner product. Let x=(2,1+i,i) and y=(2-i,2,1+2i). Compute…

A: Given:

Q: 2. * Let R' have the Euclidean inner product and let u = (1,2, – 1) and v = (-3, 1,0). Then proj,v =…

A:

Q: 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…

A:

Q: Prove that the four points P(2, ;1, 1), Q(1, 3, ; 2), R((2, 1, ; 3) and S(3, 2, 0) are coplanar

A: Given four points are  P(2,1,1)Q(1,3,2)R(2,1,3)S(3,2,0)

Q: Let R have the Euclidean inner product and let u = (1, 2, –1) and v = (3, 1,0). Then proj,u O (G…

A:

Q: In Question, compute the indicated quantity using the specified inner product.

A: Given: a0+a1x+a2x2,b0+b1x+b2x2=a0b0+a1b1+a2b2 Using the identity v=v,v. 1+x+x2=1+x+x2,1+x+x2…

Q: (u, v) is an inner product. Prove that the given statement is an identity. Prove that llu + vll =llu…

A: Prove the given statement as follows.

Q: Find an upper triangular U (not diagonal) with U2 = I which gives U = u- 1.

A: Given, U2=I

Q: Let S in Mzx2; a + 2d=0 and b-2c 0 with the standard %3D operations in M2x2 Which of the following…

A:

Q: (u, v) is an inner product. Prove that the given statement is an identity.               (u, v) =…

A:

Q: If u xv=uxw and u ‡ ♂, then u = w.

A: The given statement is: If u×v=u×w and u≠0→, then u=w

Q: 23. Prove that (xo, Yo), (x1, y1), and (x2, y2) lie on a line if and only if (х — Хо)02 — Уо) - (*з…

A:

Q: 2) Let R' real numbers, define = x,Y1 + 3x2y; + 5x3y3 for x = (x, X2, X3) and y (y,Yz, Y3). Is…

A: Given x=x1,x2,x3, y=y1,y2,y3 in ℝ3. Define x,y=x1y1+3x2y2+5x3y3

Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

A:

Q: Which elements are invertible in Z4[x,y]?

A: W need to find elements which are invertible inZ4x,y.

Q: Show that the maximum norm on C[a, b] is not induced by an inner product

A: to show that the maximum norm on Ca,b is not induced by an inner product . as we know , a norm X,·…

Q: (u, v) is an inner product. Prove that the given statement is an identity.             llu + vll2 =…

A:

Q: Let xe" and yer show that <xiy)=ytx is on imer product %3D

A: We have to show that &lt;x,y&gt; = y*x is an inner product

Q: Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is…

A: Given : The operation on ℝ2 is defined as  x, y = x1y1 - x1y2 - x2y1 + 3x2y2 ,  where x = (x1 , x2)…

Q: Let X,X2, ,X, be a ranc *(X), = 0*(1– 0)1-x; %3D

A: from above distribution is binomial  f(x)θ=θx(1-θ)1-x  ;0&lt;x&lt;1 E(x)= nθvar(x)=nθ(1-θ)…

Q: Find (a) projvu and (b) projuv. Use the Euclidean inner product.u = (1, 2, −1), v = (−1, 2, −1)

A:

Q: Show that the functions o,(x) = sin(nax²), for n = 1,2,3,..., form are orthogonal set with respect…

A: Consider the equation

Q: Let w = -4xy – 5yz – 4xz, x = st, y = et, z = t Compute (3,4) = as (3,4)%3D at

A:

Q: Find (a) projvu and (b) projuv. Use the Euclidean inner product.u = (−1, 4, −2, 3), v = (2, −1, 2,…

A: Given:

Q: Let R' have the Euclidean inner product and let (1, 2, – 1) and v = (3, 1,0). Then proj,„v = U = O…

A:

Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

A:

Q: Let R' have the Euclidean inner product and let u = (1, 2, -1) and v = (3, 1, 0). Then proj u 5 10 6…

A: Let IR3 have the Euclidean inner product space and let u=91,2,-1) and V=(3,1,0)then projuu

Q: 7. If in an inner product space, (x, u) =(x, v) for all x, show that u = v.

A:

Q: Find (a) projvu and (b) projuv. Use the Euclidean inner product.u = (5, −3, 1), v = (1, −1, 0)

A:

Q: 7. Determine whether (u,v) =3u, v, – 2u, v, is a valid inner-product function or not for R²

A:

Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

A:

Q: Let S,(x) = 2x³ – 3x² + 3x – 4, 0<x < 1, S,(x) = x³ + b,x² + c,x + d,1 sxs 2. Find b,, C1, d, that…

A:

Q: Find (u, v), ||ul|, |v||, and d(u, v) for the given inner product defined on RT. u = (1, 3, 0), v =…

A:

Q: Let ū = 2i – k and ī = i+3j + k. Compute |ū], |T], ū x ī, and Projzū.

A: Solution is here

Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: 18) (Section 9.7) Find the cross product in space u = 3i – 2 j+4k v=i-3j-k

A:

Q: Let F(x, y, 2) = (z, 1, y). a. Find divF and curlF.

A:

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

A:

Q: Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi =…

A:

Q: 2. Let W = Span {u1, u2, uz3, U4, U5, U6}, where 4 1 , U5 = Uj = U2 = u3 = ,u4 = U6 = 2 2. 4 What is…

A:

Q: In C[-1,1], with the standard inner product f(x)=ex and f(x)=e−x are orthogonal. true or false?

A: true please see explanation in step 2

Q: at x and y are orthogonal

A: Given: x and y in ℝn, x+y=x-y To show:  x and y are orthogonal in ℝn if and only if x + y = x − y

Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (-4, 0), v =…

A:

Q: If both z + w and zw are real, how to prove that either z = w (conjugate) or Im(z) = Im(w) = 0?

A:

Q: Let X=(x1,x2) and Y==(y1,y2) belongs to R^2. Verify that = 5(5x2y2+ x1y1-2x1y2-2x2y1) is an inner…

A: Definition of Inner product space Let u,v and w be vectors in a vector space V, and let c in ℝ. An…

Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on R". u = (-5, –12), v=…

A: For the given inner product

Q: Prove that Srxdr = 2[, , da, r × dr =

A:

Q: Suppose that A = SS−1, where  is a diagonalmatrix with diagonal elements λ1, λ2, . . . , λn. Show…

A:

Q: If Ø : R3 → R3 be defined as Ø (x,y,z) = (2x, 4x –y,2x+ 3yz) then show that Ø is invertible. %3D

A: Given that ϕ:R3→R3 is defined as ϕx,y,z=2x,4x-y,2x+3yz. Now, ϕ is invertible if ϕ is one-one and…