Let T/6 : R² → R² denote the linear transformation which rotates a vector in R- counterclockwise by T/6 radians. Let Tref : R² → R² be the linear transformation which across the line x1 = x2. x2 reflects a vector x = (a) Sketch a cartoon illustrating what these linear transformations do to the vector e1 = (b) Find the standard matrices of T/6 and Tref. (c) Find a nonzero vector v E R² such that Tz/6 © Tref (v) = T,ref o Ta/6(v) or explain why no such vector exists.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let T/6 : R² → R?
counterclockwise by T/6 radians. Let Tref : R² → R- be the linear transformation which
denote the linear transformation which rotates a vector in R-
across the line xj = x2.
x2
reflects a vector x =
(a) Sketch a cartoon illustrating what these linear transformations do to the vector ej =
(b) Find the standard matrices of T-/6 and
Tref-
(c) Find a nonzero vector v E R² such that
T/6 o Tref (v) = Tref o Tr/6 (v)
or explain why no such vector exists.
Transcribed Image Text:Let T/6 : R² → R? counterclockwise by T/6 radians. Let Tref : R² → R- be the linear transformation which denote the linear transformation which rotates a vector in R- across the line xj = x2. x2 reflects a vector x = (a) Sketch a cartoon illustrating what these linear transformations do to the vector ej = (b) Find the standard matrices of T-/6 and Tref- (c) Find a nonzero vector v E R² such that T/6 o Tref (v) = Tref o Tr/6 (v) or explain why no such vector exists.
Expert Solution
Step 1

Let Tπ/6: 22 denote the linear transformation which rotates a vector in 2 counterclockwise by π6 radians.

We know the counterclockwise rotation matrix by an angle θ is cosθ-sinθsinθcosθ.

Thus, 

                          Tπ/6x1x2=cosπ6-sinπ6sinπ6cosπ6x1x2Tπ/6x1x2=32-121232x1x2.

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