4. Let W be the subspace of R³ defined by the equation 2 – y + 3z = 0.(a) |Find a basis {V1,V2} of W.(b)Find a vector V3 that is orthogonal to the plane W.Explain why B = {v1, V2, V3} is a basis of R³.(d)Let T: R³ → R³ be the linear transformation given by reflection across W.Determine the B-matrix of T.Use the result of part (d) to find the matrix of T with respect to the standardbasis.

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Answered: 4. Let W be the subspace of R³ defined…

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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6. Project the vector b = (1, 4, 2) onto the plane S that contains the vectors a₁ = (1, 2, 1) and a2 = (1, 1,0). (i) Find the 3 x 3 projection matrix P onto S. (ii) Show that P2 = P. (iii) Find a vector whose projection onto S is the zero vector. (iv) Show that I - P projects onto the line perpendicular to S.
2. Consider the vectors (2, 1,0, 0], [3, 1,0, 0], [0, 0,0, 1] in R'. Find a vector v so that v, [2, 1,0, 0], [3, 1,0, 0], [0, 0,0,1] are linearly independent.
Calculate the curl of the vector = (2x + 3y)i + (y+z)j + (xy + 2zx)k
Let {uj, uz, U3, U4} be the orthogonal basis for R' given below. Write x as a sum of two vectors vand v, with v, in Span{uj, u2, Uz} and v, in Span{u4}. Enter your answers as 4x1 arrays into v_1 and v_2 . [6] 6 uz = u = Uz = -6
The position vector [3 2 1 1]lies in the YZ plane and forms 30 degree with the principal axis. Rotate the object so that it is transformed to ZX plane where it forms 45 degree with the principal axis. 3.
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).
(b) Let P, be the plane in R which contains the line in R' with vector equation 6. X1 =| 5 + and the point (3, 2, 5). (This point is not on the line.) (i) A vector equation of P, is 6 X = 5 +s 4 +t 6. 6 (Use two points on the plane to find one of the direction vectors.) (ii) An equation in general form for P2 is x+ y+ To find this equation, you need a vector n = [a, b, c] which is orthogonal to the two direction vectors that you obtained in part (i).

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4. Let W be the subspace of R³ defined by the equation 2x – y +3z = 0. (a) I Find a basis {V1,V2} of W. (b) Find a vector V3 that is orthogonal to the plane W. Explain why B = {v1, V2, V3} is a basis of R³.