Let u = (-2, 3), v = (1, −1) and A - NOTE: Give exact answers. (a) Find ||u|| relative to the inner product on R² generated by the matrix A. || ૪ || = √5 X (b) Find d(u, v) relative to the inner product on R² generated by the matrix A. d(u, v) 1 2 2 = 2 √30 X
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- Find the norm of (-4,- 1/4,3, 1/3) belong to R^4 w.r.t the Euclidean Inner Product on R^4.Let V=C³ with the standard inner product. Let x=(2,1+i,i) and y=(2-i,2,1+2i). Compute <x,y>,||x||,||y|| and ||x+y||². Then verify both Cauchy's inequality and the triangle inequality.Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi = (i − 3)/2 for i = 1, . . . , 5.
- a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish the same result without direct evaluation of the Wronskian.In C[0,1] with inner product defined by <f,g>=integral from 0 to 1 of f(x)g(x)dx, consider vectors 1 and x. (a) find the angle theta between 1 and x (b) determine the vector projection p of 1 onto x (c) verify that 1-p is orthoganal to p (d) compute ||1-p||, ||p||, ||1|| and verify that the pythagorean law is satisfiedCheck whether the following inner product <, > defined by <alpha,beta>=x1y1 + 2x1Y2 + 2x2y1 +5x2y2, where a = (x1,x2) and B = (y1,y2) is an inner product in R or not? Justify your answer. Image of question attached.
- We consider the Complex numbers $x, y, z$ with modulus 1 with the property that:$$x+y+z=0$$and the set $$A=\left\{(\alpha, \beta, \gamma) \in \mathbb{R}^{3}, \sin \alpha+\sin \beta+\sin \gamma=0\right.\hspace{0.2cm}\text{and}\cos \alpha+\cos \beta+\cos \gamma=0\}$$a) SHow that:$$x^{2}+y^{2}+z^{2}=x y+y z+z x=0$$b) For $(\alpha, \beta, \gamma) \in A$, calculate:$$\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma$$c) SHow that if $n \in \mathbb{N}^{*}$ then, for every$$(\alpha, \beta, \gamma) \in A \text {, }$$$\sin n \alpha+\sin n \beta+\sin n \gamma=0$ if and only if " $n$ " is not a multiple of 3. Please check the attached picture for details. Please write the complete solution. Thanks in advance.Write a walkthrough for the proof of Theorem 4.24 (Cauchy’s Integral Formula for circles). Provideat least three figures and elaborate on the proof given, filling in the gaps you find.A rectangle ℛ with sides a and b is divided into two parts ℛ1 and ℛ2 by an arc of a parabola that has its vertex at one corner of ℛ and passes through the opposite corner. Find the centroids of both ℛ1 and ℛ2.
- 1. Solve by Cramer’s rule 3x + y + 4z = 11 4x – 4y + 6z = 11 6x – 6y = 3 2. Find the volume of tetrahedron given the following (1, 0, 1), (0, 1, 0), (0, 0, 1), and (1, 1, 1).a) Check that the functions e^x, e^-x,coshx are solutions of y'''-y'=0 b) show directly that W(e^x, e^-x,coshx)=0 and conclude that these functions do not form a fundamental set c)Find a fundamental set of solutions by inspection. Justify your answer by computing the wronskianLet X=(x1,x2) and Y==(y1,y2) belongs to R^2. Verify that<X, Y> = 5(5x2y2+ x1y1-2x1y2-2x2y1) is an inner product on R^2