Recapitulative

 Recapitulative

We define En as the euclidean n-space of real-valued coordinates (x1, …, xn).
We can turn En into an R-vector space by defining (x1, …, xn) + (y1, …, yn) =
(x1 + y1, …, xn + yn) and k(x1, …, xn) = (kx1, …, kxn). We shall concern ourselves
specifically with E2 and E3.
i) Show that, if ? is a rotation of angle ?? around the origin (in E2) or around
an axis that passes through the origin (in E3), then ? is a linear function
ii) Show that, if M? is a matrix representation of ?, then M? is similar to
(
cos ?? – sin ??
sin ?? cos ??
)
in E2 and is similar to
?
?
1 0 0
0 cos ?? – sin ??
0 sin ?? cos ??
?
? in E3.
Deduce that |M?| = 1.
iii) Show that, if B is a basis of elements at distance 1 from the origin and
whose lines to the origin are pairwise perpendicular, then M? with respect
to B will be orthogonal.
Remark: The converse is also true, any linear function whose matrix representation
with respect to an orthogonal basis is orthogonal and of determinant 1 is
a rotation.
iv) Find the eigenvalues of M?.
v) Show that M? is similar to M?’ if and only if ?? = ??’ .
vi) Find a 3 × 3 matrix that is both a permutation matrix and a rotation
matrix, and describe its axis and angle of rotation (with respect to a basis
of your choice). The identity matrix is not an acceptable answer.
Evaluation: 2 pts each, half credit for a partially correct proof. You can refer
to previous (but not ulterior) questions even if you didn’t do them correctly.
1

We define En as the euclidean n-space of real-valued coordinates (x1, …, xn).
We can turn En into an R-vector space by defining (x1, …, xn) + (y1, …, yn) =
(x1 + y1, …, xn + yn) and k(x1, …, xn) = (kx1, …, kxn). We shall concern ourselves
specifically with E2 and E3.
i) Show that, if ? is a rotation of angle ?? around the origin (in E2) or around
an axis that passes through the origin (in E3), then ? is a linear function
ii) Show that, if M? is a matrix representation of ?, then M? is similar to
(
cos ?? – sin ??
sin ?? cos ??
)
in E2 and is similar to
?
?
1 0 0
0 cos ?? – sin ??
0 sin ?? cos ??
?
? in E3.
Deduce that |M?| = 1.
iii) Show that, if B is a basis of elements at distance 1 from the origin and
whose lines to the origin are pairwise perpendicular, then M? with respect
to B will be orthogonal.
Remark: The converse is also true, any linear function whose matrix representation
with respect to an orthogonal basis is orthogonal and of determinant 1 is
a rotation.
iv) Find the eigenvalues of M?.
v) Show that M? is similar to M?’ if and only if ?? = ??’ .
vi) Find a 3 × 3 matrix that is both a permutation matrix and a rotation
matrix, and describe its axis and angle of rotation (with respect to a basis
of your choice). The identity matrix is not an acceptable answer.
Evaluation: 2 pts each, half credit for a partially correct proof. You can refer
to previous (but not ulterior) questions even if you didn’t do them correctly.
1

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