Consider the following inner product R?. on ((T1. Yı), (T2. Y2)) = ax122 + by,Y2 If ||(1, 3)||² = 21, (1,3) 1 (-6, 3), a + b =?

Q: Find the alue of C that satisfied the theorem for the function fCx) = 3+ Jc on the %3D in terval…

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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: 1. Find a non zero unit vecor ôî that is orthogonal to u= (1,2,1) and Uz= (2,5,4) in R³.

A: Let's find non zero unit vector.

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: - Show that the norm tion t= aT+B. Use [a, b] onto [0, 1] (7) = 1, ý(7)= T, V т,

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Q: Find 3xyd A A over the triangular region D with vertices (0, 0) (2, 0), (0, 4)

A: Given ∫∫3xy dA Over region (0,0)(2,0)(0,4)

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

A: Given, U2=I

Q: c) The operation (ao + a1r + azx², bo + bịx + b,x²) = aobo + a¡b1 – azbz defined on P2 has all the…

A: We have to check whether the operation defined on P2 satisfies all the properties to be an inner…

Q: A hransfoomation is definad by Hhe axa matox A=-a b-a atb Ifois an odd imtreger, prove %3D ushere a…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

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

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Q: Let Myn2 defined by TCA) =A+A". Fünd bases for Ker GT) and Range CT).

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

A: Concept:

Q: Let UsX +Sinx Deter of X 0n [0,)where ļ Find . derivalire fa 2x+1 x²+7 te tn 6x+

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

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

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: (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: 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: 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: 4. Show that the norm on (° given by I| (21, ..., In...) |lo= sup{|7;| : j =1,2, ...} %3D can not be…

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Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: 1) Prove that Ym,n EN such thert m¢ nt = n<m using the princdple of in duction!

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Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: Show that if, in using the Laplace development, you accidentally multiply the elements of one row by…

A: A 3×3 matrix a11a12a13a21a22a23a31a32a33.

Q: : which of the following are inner product V, CR) where u = (al1,Ne] , V= on 2

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Q: Prove Theorem (AT )T = A

A: Given: Theorem, ATT=A.

Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

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Q: 0, 2 is defined by A natural cubic spline S on 5. So()12x - 0 if 1 (1) S(r)2b(x1)c(x - 1)2d(x - 1)…

A: 5.Note that, the natural cubic spline must satisfy the conditions

Q: Show that for each 37ER?, ū=, J =, the bilirear form Lūl>= 2e,Vzt U, Vz2t Uz V,t Qus Vz defines an…

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Q: Assume that u = U (x,y) iu differentiible and x =r ca Ô and y=rsin o eapress 2. entinely in termv 0…

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Q: Evaluate the integral 108 - 2y/E+ 4ydzdy, 4 where A is the parallelogram with vertices (0,0), (),…

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Q: 1. Show that the definition (x1, x2) · (Y1, Y2) = x1Y2 is not an inner product on R?.

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Q: (2), Does the funchon fl2)= 2+4E, 2€D lift a mnmal surlae Why 2 1f so, determme this mnimal susface…

A: Consider the given function f(z)=z+14. z

Q: 3. 2. If A is 2x2 matri x aud Á = A, but %3D A+ A. show that A= 0

A: The solution is :

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: 2 In C, (x, y) = xAy* is an inner product, where A = i ). Compute (x, y) -i 3 for r = (1+ i, 3i) and…

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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)

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

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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: 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: O8.D. If w, and w, are stri (x, X2) -(y Ya) = x,y,W,+Xay,W, uinlde an inner product on R'.…

A: 8.D  Let w1 and w2 are strictly positive i.e. w1,w2&gt;0.(x1,x2),(y1,y2)∈ℝ2Define the product…

Q: = 1, prove that 1. If a + u bě+ u Ć +U %3D ди ду du ди du - 2| х +y Əz

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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: In C[-1,1], with the standard inner product f(x) = e² and f(x) = e ² are orthogonal. %3D O True O…

A: Given- fx=ex and gx=e-x. To Check- If the given functions with the inner product are orthogonal or…

Q: 3,4 show that the integsal (Xy°ty%dx + 3) Independent of the path joining the peints (1,2)and 3,4).…

A: A function F is path independent if it is conservative. And if a function F is conservative then…

Q: Assume that a1, a2, ..., an E R (with n > 2) such that a1+a2++an = 1. Prove that af+az++a > 1 (Use…

A: Cauchy-Swartz Lemma : For any real numbers  and  the following inequality holds  ,

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: In C[0, 27], with the inner product -2 (f, 8) = | f(x)g(x) dx, show that the set S = {1, sin x, cos…

A: Given: Inner product defined as: &lt;f,g&gt; = ∫02πf(x)g(x)dx To show that the set S = { 1, sinx,…

Q: Compute the indicated 3 X 3 determi­nant using cofactor expansion along any row or column that seems…

A: Compute the indicated 3 x 3 determinant using cofactor expansion along any row or column that seems…