## 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