Let v =[4 2 -1]" Considering the vector space R3 provided with the inner product defined by:(a, b) = 4a,b1 + 3azb2 + azb3, where a = az2 and b = |b2| eR3lb3.azDetermine:(a) {v}+(b) orthogonal basis B1 for {v}+(c) orthogonal basis B for R3 that contains the set B1

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Answered: Let v = [4 2 -1]" Considering the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V be the vector space of functions spanned by B = {bị 2 sin r + 6 cos a, b2 = 3 sin a + 7 cos a} where a + %3D C = {ci = sin a, c2 = of coordinates matrix P ne Z is also a basis for V. Find the change 2 = cos a} where a + C+B Ex: 5 P = C-B The coordinates of a function f(x) relative to the basis B are [f(x)]B Find the coordinates of f(æ) relative to Cand find f(æ). Ex: 5 [f(x)]c = f(x) = Ex: 5 sin z + cos a
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Consider the space P₂ which is the vector space of polynomials of degree not greater than two. We define the inner product in P2 according to the formula -Lpqdz. Also, consider the following two vectors (a) Calculate the inner product (p, q) (b) Calculate the distance between vectors p and q. (c) Calculate the norm ||q||, of vector q. (p, q) (e) For vectors p and q, calculate the inner product (p - q, p + 2q). p = x² = 5x + 1, q= −2x²+x. (d) For vectors p and q, find the orthogonal projection of p on q (projp). Your answer must be a vector from P2, i.e. a polynomial of degree 2 or less.
Compute the curl of each of the following vectors: V = (yz, xz, xy) F = (x, y, z) = = ( Note: Y 2 (x² + y² +2²)³/²¹ (x² + y² + x2)³/²¹ (x² + y² + x2)³/2 • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx = x². Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {Ax, Ay, Az} . Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. • If you need to enter O, enter it in three components, as (0, 0, 0) or {0,0,0}. V x V = ▼ xr= vx= (2A, 2A, 2A₂).

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Let v = [4 2 -1]" Considering the vector space R³ provided with the inner product defined by: (a, b) = 4a¡b1 + 3azb2 + azb3, where a = a2| and b = |b2 e R3 Lb3. Determine: (a) {v}- (b) orthogonal basis B1 for {v}± (c) orthogonal basis B for R³ that contains the set B1