Next: About this document ... Up: Math 221 Fall 98 Previous: Some Matrices and Parallelograms

## 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 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 A2 ? (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:
1.
It leaves one nonzero vector fixed (i.e. ). The vector is the axis of the shear.
2.
For any vector , the difference between and is parallel to the axis .
By looking at the results of the transformation on the four parallelograms above, find the axis of the shear . Also, show algebraically that meets the definition above of a shear transformation.
f)
Look at the result of linear transformation on parallelogram P2. 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 P2. Give a description of what you see in terms of change of scale of the sides of P2. 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.

Next: About this document ... Up: Math 221 Fall 98 Previous: Some Matrices and Parallelograms
root
2002-08-21