


We can understand matrix transformations from R^{2} to R^{2} by examining them geometrically.
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?
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?]
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?)
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.



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