17. For the following questions, find a projection matrix P that projects b onto the column spaceof A, that is, Pb E col(A) and b - Pb is orthogonal to col(A).(i) A =00]0 1 000100b =1(ii) A =1-2 4-1 1147b =

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Answered: 17. For the following questions, find a…

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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10: Let W be the plane with the equation 3x-y+ 7 = 0. Find the standard matrix P for the orthogonal projection onto W. Use the following formula P = A(ATA) ¹AT, where the matrix A is constructed using any basis for Was its column vectors. Enter the first row of the matrix P (in order) into the answer box below, separated with commas. i.e., enter the values Pi1, P12, P13-
Project the vector b = (1,3,1)" onto the line through %3D a, %3D (-1,2, 2)Т, az = (2, 2, –1)", аз3 (2, —1,2)Т. Answer the following questions: 1) Find the three projection vectors p1, p2 and p3. 2) Find the projection matrices P1, P2 and P3. 3) Verify that P1+P2+P3=I. Therefore the three vectors a1, az and az are perpendicular to each other.
b. where 17. Consider a matrix A of the form A -a a? + b? v and w such that Av entries of v and w in terms of a and b). Conclude that T(x) spanned by v. = 1. Find two nonzero perpendicular vectors = i and Aw = -w (write the = Ax represents the reflection about the line L nst ni
5. Show that the vectors a = 3i-j+ 4k, b =i-3j-2k and e = 4i– 3j+2k are linearly independent. Find numbers a, B and y such that d = 2i + 3j – k can be expressed in the form d= aa + Bb + 7c.
Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4,4)} be bases for R2, and let 3 2 40 be the matrix for T: R2 R2 relative to B. 56°F A = (a) Find the transition matrix P from B' to B. P = (b) Use the matrices P and A to find [v] and [T(v)]B, where [v]B = [-5 4]T. [V] B [T(v)]B = (c) Find P-1 and A' (the matrix for T relative to B'). p-1 = 4
[10] Find the standard matrix representation of the linear transformation from R? into R² that corresponds to reflection in the line y = x followed by projection onto the y-axis.
Find the matrix A of the linear transformation T from R2 to R² that rotates any vector through an angle of 45° in the clockwise direction. A = 9-
Let the basis B = {b1, b2} and basis C = {c1, c2} where vectors b1 = (7 -2). b2 = 2 -1 , c1 = 4 1 c2= 5 2 Find the change of coordinates matrix from B to C. (b) Find the change of coordinates matrix from C to B.
). (a) Given the points (1,2), (2,6), and (4,1). Find the equation of the straight line that best fits these points. (b) Find P, the projection matrix on C(A), where A = (c) Verify that P² = P. 1 1 12 1 4
5. Determine the matrix of the transformation that first reflects points through the horizontal x-axis and then reflects points through the line y = x.
b. Let B = {T1, t2, t3}, C = {71, 72, 73}, be two orthogonal bases for R³. Show that the change-of-coordinates matrix PB>c is orthogonal.
Let B= ((1, 3), (-2,-2)) and 8' = ((-12, 0), (-4, 4)) be bases for R2, and let be the matrix for T: R2 R2 relative to B. (a) Find the transition matrix P from B' to B. p= (b) Use the matrices P and A to find [v] and [7(v)le, where [vla [-1 31. [v] = [T(v)18- 11 p-1 11 (c) Find pand A' (the matrix for 7 relative to 8). 41 41 (d) Find [7(vil bwo ways. (7)=P(7) [vila Al- 11 11

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17. For the following questions, find a projection matrix P that projects b onto the column sp- of A, that is, Pb E col(A) and b- Pb is orthogonal to col(A). (i) A = 0 0 0 1 0 0 b = (ii) A = 1 -2 4 830.0 1 -1 1 b