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.