AATA Exercises

Exercises 12.4 Exercises

1.

Prove the identity

⟨ x, y ⟩ = \frac{1}{2} [‖ x + y ‖^{2} - ‖ x ‖^{2} - ‖ y ‖^{2}] .

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2.

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Show that

O (n)

is a group.

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3.

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Prove that the following matrices are orthogonal. Are any of these matrices in

S O (n) ?

$(\begin{matrix} 1 / \sqrt{2} & - 1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 / \sqrt{2} \end{matrix})$
$(\begin{matrix} 1 / \sqrt{5} & 2 / \sqrt{5} \\ - 2 / \sqrt{5} & 1 / \sqrt{5} \end{matrix})$
$(\begin{matrix} 4 / 5 & 0 & 3 / 5 \\ - 3 / 5 & 0 & 4 / 5 \\ 0 & - 1 & 0 \end{matrix})$
$(\begin{matrix} 1 / 3 & 2 / 3 & - 2 / 3 \\ - 2 / 3 & 2 / 3 & 1 / 3 \\ 2 / 3 & 1 / 3 & 2 / 3 \end{matrix})$

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4.

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Determine the symmetry group of each of the figures below.

$There are two figures in the top row and one figure on bottom. The top left figure, a, is a rectangle. Inside the rectangle is an oval in the lower left and a solid circle in the upper right. The top right figure, c, is there intersecting circles of the same radii. The bottom middle figure, b, is a large square with the two diagonals. The midpoints of the large square are the vertices of a smaller inscribed square.$

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5.

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Let

x,

y,

and

w

be vectors in

R^{n}

and

α \in R .

Prove each of the following properties of inner products.

$⟨ x, y ⟩ = ⟨ y, x ⟩ .$
$⟨ x, y + w ⟩ = ⟨ x, y ⟩ + ⟨ x, w ⟩ .$
$⟨ α x, y ⟩ = ⟨ x, α y ⟩ = α ⟨ x, y ⟩ .$
$⟨ x, x ⟩ \geq 0$ with equality exactly when $x = 0 .$
If $⟨ x, y ⟩ = 0$ for all $x$ in $R^{n},$ then $y = 0 .$

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6.

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Verify that

E (n) = {(A, x) : A \in O (n) and x \in R^{n}}

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is a group.

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7.

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Prove that

{(2, 1), (1, 1)}

and

{(12, 5), (7, 3)}

are bases for the same lattice.

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8.

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Let

G

be a subgroup of

E (2)

and suppose that

T

is the translation subgroup of

G .

Prove that the point group of

G

is isomorphic to

G / T .

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9.

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Let

A \in S L_{2} (R)

and suppose that the vectors

x

and

y

form two sides of a parallelogram in

R^{2} .

Prove that the area of this parallelogram is the same as the area of the parallelogram with sides

A x

and

A y .

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10.

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Prove that

S O (n)

is a normal subgroup of

O (n) .

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11.

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Show that any isometry

f

R^{n}

is a one-to-one map.

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12.

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Prove or disprove: an element in

E (2)

of the form

(A, x),

where

x \neq 0,

has infinite order.

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13.

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Prove or disprove: There exists an infinite abelian subgroup of

O (n) .

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14.

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Let

x = (x_{1}, x_{2})

be a point on the unit circle in

R^{2};

that is,

x_{1}^{2} + x_{2}^{2} = 1 .

A \in O (2),

show that

A x

is also a point on the unit circle.

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15.

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Let

G

be a group with a subgroup

H

(not necessarily normal) and a normal subgroup

N .

Then

G

is a semidirect product of

N

H

$H \cap N = {id};$
$H N = G .$

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Show that each of the following is true.

$S_{3}$ is the semidirect product of $A_{3}$ by $H = {(1), (1 2)} .$
The quaternion group, $Q_{8},$ cannot be written as a semidirect product.
$E (2)$ is the semidirect product of $O (2)$ by $H,$ where $H$ consists of all translations in $R^{2} .$

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16.

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Determine which of the 17 wallpaper groups preserves the symmetry of the pattern in Figure 12.16.

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17.

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Determine which of the 17 wallpaper groups preserves the symmetry of the pattern in Figure 12.25.

A lattices of hexagons. Each hexagon is divided into three rhombuses. — 🔗
Figure 12.25. Lattice for Exercise 12.4.17

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18.

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Find the rotation group of a dodecahedron.

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19.

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For each of the 17 wallpaper groups, draw a wallpaper pattern having that group as a symmetry group.

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