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 ,, and then hitgr1 := pargrm_image_2d([vi,vj],Ak);

display(gr1, scaling=constrained);

In these questions, some parts involve looking at a computer result, while others involve a mathematical argument. We'll use the symbol to indicate questions whose answers are primarily of a computer nature and 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 with the property that is
the identity is called
*orthogonal*. Show that and meet this definition of orthogonality. (The importance of the definition is that the associated linear transformations preserve distances and angles.) **b)**- Look at the result of transformation on the
four parallelograms above. A
*reflection*of is an orthogonal transformation with the further property that it leaves one nonzero vector fixed (i.e. ) and reverses another vector (i.e. ). The vector is called the*axis*of the reflection. Print out one picture suggesting what the axis of is likely to be. Then show algebraically that vectors and as above exist for . **c)**- Look at the result of transformation on the
four parallelograms above. Can you see why the word
*rotation*is used to describe the action of ? (To explain this, relate a picture to the ordinary English meaning of the word.) Show that has no nonzero fixed vectors, i.e. for all vectors . **d)**- It can be shown that all rotations are of the form

for some angle . Show that the matrices and are both rotations. (It is in fact true that the composition of two reflections is always a rotation.) **e)**- Look at the result of transformation on the
four parallelograms above. A
*shear transformation*of is a linear transformation with the properties:- It leaves one nonzero vector fixed
(i.e.
). The vector is the
*axis*of the shear. - For any vector , the difference between and is parallel to the axis .

- It leaves one nonzero vector fixed
(i.e.
). The vector is the
**f)**- Look at the result of linear transformation on parallelogram . Explain using the definition above why this picture suggests is a shear. What is its axis ( or )?
**g)**- Look at the result of transformation on the
four parallelograms above. A
*dilation*is a linear transformation preserving angles and directions, but not necessarily distances. Show for any two vectors and , the angle between them is the same as the angle between their images and **h)**- Look at the result of transformation on parallelogram . Give a description of what you see in terms of change of scale of the sides of . Does this change of scale interpretation appear to hold for any of the other parallelograms ?
**i)**- Print out the result of transformation on one of these parallelograms, and explain the result.

One reason for talking about shears, dilations, and orthogonal
transformations is that any linear transformation of
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

This also generalizes to higher dimensions.