by Margaret Sinclair
A nice application of spreadsheets is to use them to investigate linear transformations. They take the drudgery out of matrix multiplication and allow instant graphing of both the original and the image. This unit investigates the effect of a variety of linear transformations on a unit square and a rectangle. Students record their observations in a table and conjecture relationships between characteristics of the transformation matrix and the shape, orientation and area of the image.
If we are given a square OABC with points O(0,0), A(1,0),B(1,1) and C(0,1), the general linear transformation: maps the basis vectors to the vectors represented by the columns of the matrix. That is:
Example: The transformation matrix will map the basis vector (1,0) to (3,4) and the basis vector (0,1) to (-2,1).
For any other vector we can express the mapping as follows:
The cells of a spreadsheet such as ClarisWorks can be used to enter a transformation matrix and the coordinates of a shape. The cell contents can be multiplied using matrix methods to produce the coordinates of the image shape. Both original and image can then be graphed and compared.
Setting Up the Spreadsheet
Follow the link to set up the spreadsheet for this activity.
Here is a collection of transformation matrices to use in the assignment. There is also a set of transformations available for printing.
Below is a partial table showing the headings and upper left column. There is a full tabulation sheet available for printing.
|ad-bc||ab+cd||a2+c2||b2+d2||Shape of image of square||Area of image||Orientation
c or cc
|Shape of image of rectangle||Area of image||Orientation
c or cc
- For each transformation matrix fill in the first 4 columns in the tabulation sheet. Hint: You can program the spreadsheet to do these calculations for you.
- Set up the spreadsheet to calculate and plot the image coordinates for square OABC.
- Apply each transformation to OABC and record the following:
- the shape of the image OA'B'C' (line,square, rectangle, rhombus, or general parallelogram).
- the area of the image (if applicable).
- the orientation of the vertices of the image (if the image is a line write n/a for not applicable). The orientation of the original square is counterclockwise (cc) since moving from O to A to B to C takes us in a counterclockwise direction. To check the orientation of the image, find the direction when moving from O' to A' to B' to C'.
- Set up the spreadsheet to apply the transformation to rectangle OADE where the points are O(0,0), A(1,0), D(1,3) and E(0,3).
- Record the shape, area and orientation of the image of the rectangle.
- Answer the questions below using the information in your observation chart.
- The determinant of a transformation is ad-bc. Explain any pattern you notice.
- The dot product of the vectors:
is ab+cd. What are the shapes of the images for the transformations which have ab+cd=0?
- What are the shapes of the images for which a2+c2=b2+d2? What do these two columns represent?
- What are the shapes of the images which satisfy both the conditions ab+cd = 0 and a2+c2 = b2+d2?
(Transformations satisfying these two conditions simultaneously are called similarity transformations. Similarity transformations are those for which the image figure is similar to the original figure.)
- Describe the shape and area of the images for the transformations satisfying simultaneously the conditions: a2+c2=1, b2+d2=1, ab+cd=0. Which transformations satisfy these conditions? What are the two possible values for the determinant ad-bc for these transformations?
- The orientation of the vertices O, A, B, C of the original square is counter-clockwise. How does the sign of the determinant ad-bc affect the orientation of the images?
- Calculate the ratio of the areas of the image to the original. Identify any pattern you notice.
- If the linear transformation is an isometry, then the lengths of vectors and angles betweeen vectors remain unchanged under the transformation. The image of the unit square in this case is another square of the same size.
Show that the possible matrices for an isometry are:
where x is some real number.
Related sites you may wish to visit:
[Vectors, a unit by Gene Klotz]
[Graphing with Spreadsheets - Suzanne Alejandre]
[Ask Dr. Math]
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