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.

Answered: Image /qna-images/answer/a7190eef-3812-4520-afc8-455c023b3c71.jpg

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.
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
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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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.