Let Ps have the inner product given by evaluation at-2,-1, 0,1, and 2. Let p(t) = t2- 2t and q(t) = t² +4. Find (q, q)

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: Which of the sets is orthoponal under the given inner product on C[O, T]? = S Puigenldx I. {1, sinx…

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Q: Q2= A) for E= 2xĭ-3Y3+K %3D prove That divc curl È)=0 %3D

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Q: Let f(t) = (t – 2)*(t+1)² be the characteristic polynomial for a linear operator T. Obtain all…

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

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Q: Find the SPI form of the Boolean expression whose Karnaugh map is below. yz _yz yZ ỹz 1 1 | 1 1 wx…

A: From the given table, the input value of Karnaugh map is…

Q: Find the Wronskian of a given set {yı, y2} of solutions of 7. (1 x2)y"-2xy' (a 1)y0, _

A: Consider the given equation.

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: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

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Q: Consider the following inner product: (S, 9) = /.2 f(x)g(x) VI dx. Show that the set of functions…

A: We will prove the given statement.

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 y1 = 9cos(2x) and y2 = 2cos2 (x) – 2sin? (x). Calculate the Wronskian, W(y1, Y2). Give your…

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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) = 3x? +…

A: Consider P2 (polynomials of degree atmost 2) with the inner product defined as follows, p, q=∫-11…

Q: (a) Let f(r, y) = -2r*y² – In(y²). Compute fyzz (b) Let g(r, y, 2) = 1(3y* – 52²). Compute

A: (a) Given function is fx,y=−2x5y2−lny2 We have to find the fyxx. First differentiate…

Q: Let yk = k² and Zk = 2k|k|. Are the signals {yk } and {Zk } inearly independent? Evaluate the…

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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: Q.4.(a). Show that the following functions is not an inner product on R3, where u = (u, U2, U3) and…

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Q: Let f(x, y, z) = In(x² + z) + y²e®%. Find fzym. Zyx•

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Q: 2. In A ABC, X is the centroid. A A If CW = 15, find cx and XW. W, B. If BX = 8, find BY and XY.

A: we know that  centroid is located 23 of the distance from the vertex along the segment that connects…

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: 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: 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: 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: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: find an antiderivative Fon of the function fox)=5sinX such that FCO)=1.

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Q: Use Cramer's rule to solve for y without solving for x, z, and w. 5 x + y + z + W = -7 5x + 3y - z +…

A: Here we have to use cramer rule for solving the system of linear equations

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

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Q: Calculate A and B so that t B %3D 1+1 (1-z²)

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Q: Consider P2 with the standard inner product. Then which one of the foilowWing pair is orthogonal a)…

A: We are given: Inner product space P2 with standard inner product To find: which pair is orthogonal

Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

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: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, ||g||, and d(f, g) for the inner…

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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: Let, T: R3 → R³;T(x, y, z) = (2x + y, y – z,2y + 4z) Find the rank and nullity of T.

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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: Q6. Let C[0,1] have the inner product < f,g 2 S,"f(x)g(x)dx and let fn= cos nx (n=0,1,2,3,....).…

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Q: Let C be the rectangle in the xy-plane with vertices (0,0), (1,0), (0, 2) and (1, 2), and let F(r,…

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Q: Let A € R"Xm, n > m. a) Use the SVD of A to deduce the SVD of AT A. b) If m = n and A is full-rank,…

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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: Find the Jacobian of following functions f(x,y, z) = 3xy – yz + 2y z2 - 5xyz3 g(x,y, z) = xyz -…

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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: Find (u, v), ||u|, |v |, and d(u, v) for the given inner product defined on R". u = (-5, 4), v = (0,…

A: We have given vectors u and v,  And we have to find a), b), c) , d)

Q: Let z = f(x, y) and suppose x = 2u – 3v and y = 5u + 6v. Use the chain rule to express dz in terms…

A: x=2u-3vy=5u+6v Now we apply the chain rule  ∂z∂u=∂z∂x·∂x∂u+∂z∂y·∂y∂u Take derivative of x  and y…

Q: Use Gramer's Rule to Find K,y and Z -y-22 = -8 %3D + 32 = 2 %3D 7X +y+Z = 0

A: As per our guidelines we are supposed to answer only one question at a time. So submit the other…

Q: Let z = F(x , y) = xy2 + - , where x = u2 + v2 - 4 and y = uv . az Use the chain rule to find at (u…

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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…

Q: For p(t) = a2t2 + a1t + a0 and r(t) = b2t2 + b1t + b0 in P2[t], define := a2b2 + a1b1 + 3a0b0. a.)…

A: a.&lt;p(t),r(t)&gt;:=a2b2+a1b1+a0b0&lt;p(t),r(t)&gt; is an inner product if it satisfies three…