Questions

As you do the following questions, we'd like you to look at a fair number of results of

gr1 := pargrm_image_2d([vi,vj],Ak);
display(gr1, scaling=constrained);

but don't print them all. Also note that once you have two lines like the above in your Maple session, you can just edit the indices i,j,k and then hit ENTER again to see the next set of pictures.

In these questions, some parts involve looking at a computer result, while others involve a mathematical argument. We'll use the symbol $^\dag$ to indicate questions whose answers are primarily of a computer nature and $^\ast$ for questions requesting other arguments. Some parts of the questions, are providing defintions and explanations; you don't need to answer those of course!

a)

A square matrix ${\bf A}$ with the property that ${\bf A^t A}$ is the identity is called orthogonal. Show$^\dag$ that ${\bf A1}$ and ${\bf A2}$ meet this definition of orthogonality. (The importance of the definition is that the associated linear transformations preserve distances and angles.)

b)

Look$^\dag$ at the result of transformation ${\bf A1}$ on the four parallelograms above. A reflection of ${\bf R^2}$ is an orthogonal transformation ${\bf T}$ with the further property that it leaves one nonzero vector ${\bf u1}$ fixed (i.e. ${\bf T(u1)}={\bf u1}$) and reverses another vector ${\bf u2}$ (i.e. ${\bf T(u2)=-u2}$). The vector ${\bf u1}$ is called the axis of the reflection. Print$^\dag$ out one picture suggesting what the axis of ${\bf A1}$ is likely to be. Then show$^\ast$ algebraically that vectors ${\bf u1}$ and ${\bf u2}$ as above exist for ${\bf A1}$.

c)

Look$^\dag$ at the result of transformation ${\bf A2}$ on the four parallelograms above. Can you see why$^\dag$ the word rotation is used to describe the action of A2 ? (To explain this, relate a picture to the ordinary English meaning of the word.) Show$^\ast$ that ${\bf A2}$ has no nonzero fixed vectors, i.e. ${\bf A2(u) \neq u}$ for all vectors ${\bf u} \neq 0$.

d)

It can be shown that all rotations are of the form

$${\bf R} = \left(\begin{array}{rr} cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array}\right),$$

for some angle $\theta$. Show$^\ast$ that the matrices ${\bf A2}$ and ${\bf A1^2}$ are both rotations. (It is in fact true that the composition of two reflections is always a rotation.)

e)

Look$^\dag$ at the result of transformation ${\bf A3}$ on the four parallelograms above. A shear transformation of ${\bf R^2}$ is a linear transformation with the properties:

1.

It leaves one nonzero vector ${\bf u1}$ fixed (i.e. ${\bf T(u1)}={\bf u1}$). The vector ${\bf u1}$ is the axis of the shear.

2.

For any vector ${\bf u2}$, the difference between ${\bf T(u2)}$ and ${\bf u2}$ is parallel to the axis ${\bf u1}$.

By looking at the results of the transformation ${\bf A3}$ on the four parallelograms above, find$^\dag$ the axis of the shear ${\bf A3}$. Also, show algebraically$^\ast$ that ${\bf A3}$ meets the definition above of a shear transformation.

f)

Look$^\dag$ at the result of linear transformation ${\bf A4}$ on parallelogram P2. Explain using the definition above$^\ast$ why this picture suggests ${\bf A4}$ is a shear. What is its axis ($^\dag$ or $^\ast$)?

g)

Look$^\dag$ at the result of transformation ${\bf A6}$ on the four parallelograms above. A dilation is a linear transformation preserving angles and directions, but not necessarily distances. Show$^\ast$ for any two vectors ${\bf u1}$ and ${\bf u2}$, the angle between them is the same as the angle between their images ${\bf A6(u1)}$ and ${\bf A6(u2)}$

h)

Look$^\dag$ at the result of transformation ${\bf A5}$ on parallelogram P2. Give$^\dag$ a description of what you see in terms of change of scale of the sides of P2. Does$^\dag$ this change of scale interpretation appear to hold for any of the other parallelograms ?

i)

Print$^\dag$ out the result of transformation ${\bf A7}$ on one of these parallelograms, and explain$^\ast$ the result.

One reason for talking about shears, dilations, and orthogonal transformations is that any linear transformation of ${\bf R^2}$ can be expressed as a composition of one each of these together with one more basic transformation, a strain. A typical strain would be given by the matrix

$$\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right)$$

This also generalizes to higher dimensions.