Consider the linear transformation T: R³ → R³ that rotates vectors 90 degrees around the line (x, y, z) = (t, 2t, 3t). (The orientation of this rotation is the one given by the direction vector w = (1, 2, 3) and the right-hand rule; we aren't going to worry too much about the orientation here.) Find the matrix representation for this transformation as follows: (a) Find the matrix representation R of a rotation around the z-axis by 90 degrees. (b) The line (t, 2t, 3t) is orthogonal to the plane x+2y+3z = 0. Find a basis (v₁, v₂) for the plane x + 2y + 3z = 0 in R³. (c) Find the matrix representation of the transformation S such that S(vi) = ei, S(v₂) = ₂, S(w) = €3 (Think carefully about how to find this!)
Consider the linear transformation T: R³ → R³ that rotates vectors 90 degrees around the line (x, y, z) = (t, 2t, 3t). (The orientation of this rotation is the one given by the direction vector w = (1, 2, 3) and the right-hand rule; we aren't going to worry too much about the orientation here.) Find the matrix representation for this transformation as follows: (a) Find the matrix representation R of a rotation around the z-axis by 90 degrees. (b) The line (t, 2t, 3t) is orthogonal to the plane x+2y+3z = 0. Find a basis (v₁, v₂) for the plane x + 2y + 3z = 0 in R³. (c) Find the matrix representation of the transformation S such that S(vi) = ei, S(v₂) = ₂, S(w) = €3 (Think carefully about how to find this!)
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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Transcribed Image Text:(5) Consider the linear transformation T: R³ R³ that rotates vectors 90 degrees
around the line (x, y, z) = (t, 2t, 3t). (The orientation of this rotation is the one
given by the direction vector w = (1,2,3) and the right-hand rule; we aren't going to
worry too much about the orientation here.) Find the matrix representation for this
transformation as follows:
(a) Find the matrix representation R of a rotation around the z-axis by 90 degrees.
(b) The line (t, 2t, 3t) is orthogonal to the plane x+2y+3z = 0. Find a basis (v₁, ₂)
for the plane x + 2y + 3z = 0 in R³.
Find the matrix representation of the transformation S such that
S(vi) = ei, S(v₂) = 2, S(W) = €3
(Think carefully about how to find this!)
(d) Geometrically, describe the linear transformation
SRS-1
(e) Set T = SRS-¹ and check that it performs the rotation appropriately. (In
particular, check that Tw=w and that Tuj is perpendicular to v₁. Technically,
there are some issues to be wary of with regards to orientation, but we'll gloss
over that.)
(f) Is T diagonalizable? How do you know this without computing eigenvectors?
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