Linear Transformations, Part 3

Linear Transformations

Part 3: The Geometry of 2 x 2 Matrix Maps

We can understand matrix transformations from R² to R² by examining them geometrically.

In your helper application worksheet, you will find commands to plot a grid of points, define a 2 by 2 matrix, and plot the transformation of the grid points under the matrix transformation. Execute these commands, and describe the transformation defined by the matrix T. Which points in the unit square [0,1] x [0,1] are mapped to themselves under this transformation?

Now let T be the matrix

Describe the transformation defined by this matrix. Which points in the unit square [0,1] x [0,1] are mapped to themselves under this transformation?

The transformations defined by the matrices

and

both take the unit square to the fourth quadrant. How are the transformations different? [Hint: Where does each transformation take the vector (1,0)? What points are mapped to themselves under each transformation?]

Find a matrix which defines a linear transformation that reflects the unit square to the square with vertices (0,0), (0,-1), (-1,-1), and (-1,0). Can you find more than one? Explain.

Matrices of the form

and

are called contraction and expansion matrices. Try several values for k between 0 and 3 to see how the expansions and contractions work. [Note that you may need to adjust the view of your grid plots if k is large.] Carefully describe the geometry of the transformations defined by each of the matrices

and

(When is the transformation a contraction? When is it an expansion? How do the transformations defined by the two matrices differ?)

Experiment with different values of k in the shear matrices defined by

and

Try both positive and negative values for k. Explain why matrices of the first form are known as horizontal shears and matrices of the second form are known as vertical shears.

Find a matrix that acts as a dilation by a factor of 2. That is, the transformation defined by the matrix stretches the unit square to a square with vertices (0,0), (2,0), (2,2), and (0,2). What is the general form for a dilation matrix? Could you similarly define some sort of contraction matrices? Explain.