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(Solved): A shear along a hyperplane: Let n be a fixed unit vector, and let , the hyperplane that is the ortho ...
A shear along a hyperplane:
Let n be a fixed unit vector, and let , the hyperplane that is the orthogonal complement of the line spanned by n.
Given a fixed, non-zero vector hinH, a shear by h that fixes H is the linear transformation S:R^(n)->R^(n) that sends n to n+h, and
leaves all vectors in H unchanged.
That is,
S(n)=n+h, and ,S(v)=v for any vinH
Problem: let H be the orthogonal complement of the line spanned by n=[[(3)/(5)],[(4)/(5)]], and let h=[[-8],[6]], a vector that belongs to H. Find
the standard matrix which represents the transformation S:R^(2)->R^(2) that is a shear by h that fixes H.
Steps:
Verify that n is a unit vector and that h is orthogonal to nH ◻={n,h} is a basis for R^(2).
Determine [S]_(◻larr◻)=[[S(n)]_(◻)[S(h)]_(◻)].
Determine [S]=P_(◻larr◻)[S]_(◻larr◻)P_(◻larr◻)
Answer
[S]=[[◻,◻],[◻,],[◻,]]